You should be used to it on the internet by now, but just in case, we're going to be talking THE FORCE AWAKENS SPOILERS here.
1. Was the New Republic completely destroyed by the Starkiller?
If you thought the New Republic was dead and gone, well buddy... That remains to be seen, actually. The Hosnian System that gets axed by Starkiller Base was indeed home to the New Republic senate at the time of The Force Awakens. What the movie doesn't make clear is that this isn't the entirety of the New Republic.
That would have been a huge bummer. Han, Luke, Leia, Lando, and Chewie fight for three movies and endless comic book and TV tie-ins to come only to have all of their work undone by JJ Abrams and a crack CGI team? No. There are definitely other worlds out there free from the First Order and the lawlessness of the Outer Rim.
With the death of Emperor Palpatine -- and the Rebel Alliance's subsequent PR boost -- the New Republic wasn't actually formed around the Hosnian system at all. It was established on Chandrila, home world of the senator-turned-rebel-leader-turned-chancellor Mon Mothma.
When the New Republic was formed she essentially took over Palpatine's pre-emperor role. Logically, a lot of people thought her motherland would make a fine capital. But even that didn't last. Eventually, the senate decided to rotate their capital from system to system on a vote every few years. Like the Olympics, but with (hopefully) less bribery. When General Hux, Supreme Leader Snoke, and Kylo Ren egged the Republic's house with a solar laser they only destroyed the government's leadership infrastructure, and an unknown percentage of its space fleet.
So... basically the First Order did exactly to the former Rebel Alliance what had been done to them. They cut off the head of their government, and crippled their military power. There are still plenty of New Republic worlds out there, but if what happened to the Empire after Return of the Jedi is any indication they're in for a bad time.
Too bad nobody saved up any Bothan spies to warn them about this fully armed and operational battle station.
2. How did Kylo Ren get Darth Vader's Helmet?
One of the more iconic shots in Star Wars: The Force Awakens is that of a pensive Kylo Ren beseeching one very nuked Darth Vader helmet for advice. The general meltiness, we get -- Luke burned Anakin's suit along with this rest of his daddy issues on Endor. But how did the man once known as Ben come to own his grandfather's headwear?
Well, while it's not entirely confirmed there is one substantial theory floating around based on an in-canon story called "The Perfect Weapon." It stars Bazine Netal, the snitch seen lounging in Maz Kanata's cantina. You know, the one who rats on Finn, Rey, Han, and Chewie to the First Order?
In this story Bazine meets up with an ex-stormtrooper. An individual who it just so happens survived whatever the hell it is Ewoks did to the owners of all those helmets they had on spikes at the end of Return of the Jedi. The equally canon novel Star Wars: Aftermath describes this process in vague yet terrifying detail, actually, as well as setting up the idea that some Imperials could have disguised themselves as rebels to escape capture, execution, and Ewok evisceration following the decisive battle.
The retired trooper passes off a package to the merc/informant which is never seen, to a client that is never referred to by name. While that's not a lot to go on, the mystery surrounding these two elements would imply that they were important figures. Say Kylo Ren and his favorite paperweight, for example. The trooper would have been in prime position to nab the plastic, as it seems unlikely Luke would have carried off the charred remains himself.
Without confirmation it's really just a theory, but it's just too juicy to ignore. Especially since the short story was part of the "Journey to Star Wars: The Force Awakens" line of tie-ins, which were all meant to establish something important for Episode VII.
Either way, it looks like Disney has wasted no time in setting up enigmatic no-names throughout their kick on Star Wars. George Lucas would be proud, if he could tear himself away from eating $100 bill sandwiches.