In a room that I apparently need to kill every mob to open. Do that and gained a level. Now I can't exit the room or progress.

I've had the same issue on my first run and that has put me off playing any more of the game.
It first happened in a common room like you, where the doors wouldn't open after clearing the ennemies. I saved and loaded and went in again only to face the same problem.
Then, I decided to skip that room and head to the boss room and got another failed trigger there too. I entered, the door closed and nothing happened. No boss, no dialog, no way to exit and most importantly no way to progress the story. Save/load 4 more times and got the same issue every time.

It seems they are aware of such issues looking at their support thread on the Steam forums, and other players have reported the same thing.
Unfortunately there is no solution yet except "delete your run and start over" which in a 3-4 hour run is quite disheartening.
I sent them an email with screenshots to explain the issue but haven't heard back yet.

I talked with Bidds (the dev) on twitter and sent him my save files.
Just hoping he can figure that out and patch it so it will not happen again.
Try zipping your files and send it to support@onebitbeyond.com
IMO the more data the better.

The game is still buggy as hell. I have uninstalled it for now and hope that they will fix the bugs.