All of the above are good suggestions. I'll re-iterate with a little first hand experience from an owner of a '74 Overlander which has had it's fair share of electrical gremlins.
There is a 50A (DC) fuse mounted to the wall just to the left of the Univolt. Make sure this is clean and maybe even install a new fuse for the heck of it. This one thing caused me some havoc for a while before discovering it. I found out it was the problem when I lost all 12V
power while camping. I spun the fuse in it's holder and all of a sudden, lights came on! I cleaned things up, put in a new fuse, and no issues since.
Also, I've had a lot of issues with a good connection at the battery. Get a battery terminal brush-cleaner and ream on the connections until they shine, then but back together nice and snug. Do this every year.
I have an IOTA converter in mine and it has been pretty good. I have a Univolt still in my other Airstream and it hasn't given me any issues. But, I don't leave it plugged in 100% of the time, either.
Your second question perhaps needs more detail. Do you want to know if your truck will charge a dead battery? Well, eventually maybe. But typically if a battery is dead-dead, it needs to be taken out and cycled and all sorts of fun stuff while monitoring it with a voltmeter. It may not recover. Your truck won't do much in this case. If you're asking if 12V from your truck is being supplied to the trailer when connected, that answer is yes, or at least it should be if all the wiring is correct.
If you know you are always going to have 120V AC power when you're camping, then I wouldn't worry much about a battery and keeping it charged and such. As long as the Univolt is providing 12V DC when you're plugged in to 120V AC, then everything in your trailer should be functional when you're camping. If you get into boondocking then the battery health and keeping it charged is a whole new story. I would agree with unplugging your trailer from 120V AC power when you're not camping in it. A battery disconnect switch might be a good addition in this case, too. Or, manually disconnect the battery as mentioned before in this thread.