Okay, my two cents:
I think we have to consider the format of how the next book will play out. Unlike the first two, I doubt it will be 95% flashbacks and storytelling. Most likely, the first 60% will be Kvothe catching the readers up to the current day. The other 40% will be Kvothe having regained his powers (somehow, possibly Bast bringing Denna to the Inn in a last ditch attempt) and Kvothe going after the rest of the Chandrian, however many still survive.
Kvothe obviously (whether on purpose, or accidentally) started the war that rages the lands in the present day and he holds himself SO personally responsible for his actions (Most likely he couldn't stomach seeing so many die due to his actions) that he locks his powers away in that chest with a name of some sort.
Near the end of the TWMF Kvothe makes an attempt to open the chest, but can't. This is interesting because it leads me to believe that HE has the ability to open it himself, but still can't get through his own mental barriers to actually DO it. Hence why he makes an ATTEMPT to fight off the soldiers hired by Bast, but still gets his ass handed to him anyways. A part of Kvothe OBVIOUSLY wants his powers back, but a larger portion of him, (His sleeping mind perhaps) is still keeping him from taking that next step in his recovery from trauma.
I believe Bast feels that he needs to up his efforts another notch and does try to bring some surviving character or characters from the story to him to help wake him up. I wouldn't be surprised to see Bast suddenly up and leave when the third book begins and go off looking for something or someone to help while Kvothe continues telling his tale to Chronicler.
It's also possibly that Bast goes in search for Felurian as his 'ace in the hole'. Letting her know where Kvothe is. If anyone outside of Denna could 'wake up Kvothe' from his trauma, I think it would be Felurian.
As for the 'Doors of Stone', I don't think any of the Chandrian are held or imprisoned by the doors. It's possible that it is a prison though for REAL names and that somehow those doors open, unleashing chaos around the world as Kvothe or whomever suddenly has unthinkable power over those names.
Despite where the second book leaves off, I do strongly believe that the 3rd book ends with a showdown between Haliax and Kvothe ala 'Voldemort VS. Potter'. Whatever crimes Kvothe may have done himself, he would want the ultimate redemption, kill Haliax and bring peace to the land.
Also, I don't think Master Ash is Cinder. Cinder is 'otherworldly' obviously, and I doubt that he would have a walking stick after being injured by an arrow. The only reason Cinder flees the bandit camp is because of secrecy. I believe Cinder could have wiped out the party had he not tried to be hidden and kept things secretive.
It's also possible that the box Lady Lackless owns holds the true name of someone very important, most likely Haliax/Lanre himself. We're obviously not done with that box of hers, and the age of the box would coincide with some of the legends and tales we've heard about Lanre.
Sorry if I rambled a little. Also, I apologize for any spelling errors, I listened to the audiobooks, so I didn't see any of these names in print yet.