Obviously, this post contains DETAILED SPOILERS FOR THE FILM AFTER THE (HYPERSPACE) JUMP:
Oh my Yoda! They killed Coruscant!
Chill! That wasn't Coruscant. The planet destroyed by Starkiller Base was Hosnian Prime, the current assembly world for the New Republic. This is confirmed in (easy-to-miss) dialogue later in the film. This is also the most misleading section of the movie, as up to this point the First Order simply refers to "the Republic" and Hosnian Prime looks a lot like Coruscant.
Whither the Republic?
It appears that the New Republic/Galactic Republic is considerably smaller and less powerful than either the Old Republic or the Galactic Empire. The New Republic seems to span a much smaller number of systems and has less military power. The Star Wars galaxy at this time in fact seems to consist of numerous independent factions and worlds and a few big ones, such as the Republic (descended from the Rebel Alliance) and the First Order (descended from the Empire). I think we can also safely assume that the Hutts remain a power in certain areas of the galaxy.
Member-species of the New Republic seem to include the Mon Calamari (such as Admiral Ackbar), the Sullustans and the Bothans, as well as human worlds like Chandrila (Mon Mothma's homeworld) and Hosnian Prime. A key difference to the Old Republic is that the New Republic does not have a fixed capital. Instead the capital periodically moves from member world to member world by vote.
On that basis, Coruscant is an important member-world of the Republic (assuming it is part of it) and probably still the most populous, but it isn't the capital.
The impression gathered from the film is that the Republic and the First Order may be more comparable in strength and on an equal footing to one another than the Rebels and Empire were. A full-scale, all-out and mutually ruinous war between the two sides, following just a few decades after the Galactic Civil War and the Clone Wars, is therefore beset avoided. The Resistance is a small, plausibly-deniable, self-sufficient force moving around behind First Order lines and taking the fight to them with the tacit (but not public) support of the Republic.
The destruction of the Hosnian system and the Republic's main fleet may change this, but the loss of Starkiller Base may have also inflicted a substantial defeat on the Order, forcing both sides to regroup.
That's no moon-sized battle station...
Starkiller Base appears to have originally been a small planet, converted over the course of decades by the First Order so it could house a weapon of tremendous power. This weapon is a hyperlight plasma/laser weapon which "powers up" by draining plasma from the nearby star and then releasing it through hyperspace to hit the target. This enables Starkiller Base to hit any planet in the galaxy without having to move itself.
Hang on, so does that kill the star?
No. The base draws plasma from the star's corona and then releases it. The star itself is unharmed and returns to normal after a few hours, allowing the weapon to be fired again. Why this makes the star go dim is unknown.
How did Han Solo and everyone see Hosnian Prime blowing up in the sky?
J.J. Abrams has absolutely zero conception of how big space is (see also the destruction of Vulcan in the 2009 Star Trek movie, which was nonsensically visible from the surface of the planet Spock is on). An easy answer would be that Takodana (Maz Kanata's planet) is simply in the Hosnian system, but the official supplementary materials confirm that Takodana, Hosnian and the Starkiller planet are actually scattered across the galaxy from one another. Light from the destruction from Hosnian Prime should take tens of thousands of years to reach Takodana. You can chalk this one up to simple silliness.
Who is Max von Sydow's character?
That's Lor San Tekka, apparently a "vicar in the Church of the Force". I have no idea what any of that means.
More specifically, there's a good chance that a young version of Tekka will appear in either Rogue One or the "Young Han Solo" movie, and we will learn more about him there.
One theory is that Tekka may have been playing a similar role to Obi-Wan in the original trilogy: keeping an eye on Rey from a distance on behalf of someone else (Luke?).
How did Maz Kanata get Luke Skywalker's lightsabre?
The lightsabre was lost when Darth Vader cut off Luke's hand in their duel in Cloud City in The Empire Strikes Back. Fans assumed that the sabre (and hand) fell through a vent and was lost in the gas giant's clouds. However, this is not necessary the case: the bottom of Cloud City is actually enclosed, Luke had to fall through several vents and secondary tubes to tumble out of the bottom of the city. There's no reason that the lightsabre could have been later found (by the Imperial troops which seized the city or by others later on). How it got from Cloud City to Kanata is a different story, and one I suspect will be expanded upon in later episodes or supporting materials.
Is it a bit implausible that Han and Chewie found the Falcon so fast?
The speed is implausible, but blink-and-miss-it dialogue suggested that Han had ways of tracking the ship. As soon as it powered up and left Jakku, it sent a signal Han was able to follow.
Given Han's paranoia about losing the ship to Jabba (or anyone else) in the original trilogy, it's not implausible he had ways of finding the ship if it was captured or salvaged. He just didn't count on it (mostly) being left inactive on the surface of a planet for three decades.
How did the Falcon do so well after 30 years in a junkyard when it was falling apart in the original trilogy?
According to Rey, the guy who salvaged the Falcon actually maintained it over the years and was trying to get it flying effectively. It even sounds like Rey had been allowed to work on it and take it on brief test flights.
It's also worth remembering that Han's problems weren't so much down to the Falcon's age or condition, but more down to jury-rigging and overloading systems to make the ship fly much faster than it should have been able. This introduced a lot of problems a standard YT-1300 never would have encountered. Rey's employer may have resolvedout some of these issues with the addition of more advanced technology in the meantime.
What happened to the clones?
In this movie it is mentioned that stormtroopers are no longer all clones, but instead are now indoctrinated and trained in the First Order from birth, when they are removed from their parents and raised in an atmosphere of total discipline and control.
The reason for this is likely that continuing to use clones was just so insanely expensive. The Kamino Clonemasters charged a lot of money for their services during the Clone Wars. Secondary sources also suggest these facilities were destroyed during the war, and rebuilding them may have been prohibitively costly. The development of the clone army during the wars may have been more useful for stealth purposes (building an initial army of three million troops in total isolation from the rest of the galaxy) than being an actually viable way of maintaining military forces.
Once the need for secrecy was done and the Emperor's New Order established, it was actually more cost-effective to simply train ordinary people as stormtroopers.
How did the untrained Finn and Rey hold off the more experienced Kylo Ren in combat?
This is a good question, since at first glance Ren should have made mincemeat of them both. However, on further reflection this is not necessarily the case. The art of lightsabre combat clearly deteriorated massively after the Jedi Purge, not helped by Obi-Wan's advancing age and the fact that Anakin/Vader's combat skills were diminished by him being in the life support suit. Luke didn't have anyone to train with apart (presumably) from remotes. Also, given Yoda's teaching methods apparently focused on control and philosophy over lightsabre combat, it's likely Luke would have employed similar training methods himself. Therefore, Ren may simply have not gotten much training in lightsabre combat, certainly nothing like what the Jedi or Sith employed from the Clone Wars era.
In addition, Ren was injured by Chewbacca's bowcaster and is clearly shown to be both constantly bleeding and having to concentrate to hold off the pain. This distracts him constantly during the fight.
On the reverse side, the Stormtroopers are shown to have training in melee combat which Finn would have likely received as well, so he's not completely inexperienced in it. Rey doing so well is a little more fanciful, but again she is shown to be adept at using her staff in fights on Jakku. More to the point, and something that is easy to miss, is that she appears to be significantly stronger in the Force than Ren. Although strength in the Force cannot entirely make up for a lack of actual sabre training, it has been shown to help a lot. Combined with Ren's injury and the short duration of the fight, it is at least reasonable that Rey would have been able to hold her own.
Who are Rey's parents?
Clearly Rey was left on Jakku deliberately for some reason. She is convinced they are coming back but no indication is given as to why she should believe this. They were also probably Force-wielders, given her strength in the Force.
She does not appear to be Han and Leia's child: they never mention another child and there seems to be no reason as to why they would have abandoned her. She is also too old to be Ren's daughter.
One suggestion is that she is Luke's daughter. This is possible if we accept that Luke would have been terrified after what happened to Ren, so he put Rey somewhere safe. However, dumping her on a desert planet and hoping for the best seems completely illogical. Homing her with responsible, loving parents and hoping she never developed Force skills would have made more sense.
On that basis, Rey's parents are probably someone and somewhere else. One theory is that Rey was a "youngling", given to Luke to train in the Force when very young. Based on their relative ages, it's possible she was only in Luke's care briefly before Kylo Ren flipped out and betrayed Luke. Luke could have then hidden Rey on Jakku, possibly with Lor San Tekka to keep a distant but constant eye on her.
The real answer is that he was supposed to die on Jakku, but J.J. Arams liked Oscar Isaac so much that he kept him around. This required some fancy footwork to explain his survival, however. Most likely, the story he gives (he got to a settlement and called for help) is true and that's it.
Some fans have suggested that Poe did die and was replaced by a clone or shapeshifter, although it seems highly unlikely that this copy would retain Poe's badass piloting skills and would then help in destroying the Starkiller. On that basis, I think we can assume that his escape story is true, if a bit implausible.
So, what is Kylo Ren's deal?
Kylo Ren - real name Ben Solo - is the son of Han Solo and Leia Organa. From his apparent age (and his actor's age) he was likely born only a year or two after Return of the Jedi. At a young age, Luke agreed to train him in the ways of the Force but Ben seems to have gone off the rails, inspired both by tales of his grandfather (Darth Vader, Leia's father) and influenced by Snoke, the Supreme Leader of the First Order. Ren betrayed Luke and his fellow students to the Knights of Ren, company of Dark Side-wielding warriors. Ren then joined the First Order openly.
Despite his betrayal, Ren remains conflicted by it and often speaks of being "tempted" by the Light Side of the Force.
Who the heck is Snoke?
Snoke is the Supreme Leader of the First Order. He is very old, apparently non-human and appears to be a wielder of the Dark Side of the Force. Whatever his origins, it appears he emerged from obscurity to seize control of part of the Empire as it was falling after the Battle of Endor. He maintained control of this part of the Empire and turned it into the First Order, even as the rest collapsed, surrendered or broke away.
Apparently Snoke likes to use a hologram system to enlarge his physical appearance. Although still larger than human norms, his "real" size is not as big as the hologram. One of the supporting books suggests he is more like 7 feet tall than 25.
What was going on with R2-D2 and Luke?
R2-D2 spends most of the film powered down and Threepio is convinced that he may never function again. However, at the end of the film Artoo suddenly powers up and provides the key to the location of Luke. Why and how?
Interestingly, this happens when Rey is at the Resistance base for the first time. (she was imprisoned on Starkiller Base earlier on when Finn was at the Resistance base). If you buy the "youngling" theory above, it may be that Luke programmed Artoo to give up his location when another of his students was brought into Artoo's presence. This may also explain Luke's conflicted appearance at the end of the film when Rey shows up.
The screenwriters also have another explanation: Artoo actually has a map of the galaxy containing areas of interest to the Emperor and Vader, downloaded when he hacked into the Death Star computer (in A New Hope). This allowed him to combine the information with BB-8's to locate Luke at the old Jedi Temple.
Star Wars: Rogue One is released next December and will a self-contained action film set just before the events of A New Hope. Episode VIII, which will continue the story, will be released in May 2017...less than eighteen months from now.