I started a new game, in Morrowind. After a short while, the doors to the Census and Excise Offices became locked, with me unable to enter.
I wanted to turn in the information and money on the dead tax collector, Processus Vitellius. At some point, the door unlocked. I went all the way through, only to find, after turning in what I needed to, the door locked on me. And I couldn't "wait" to see if it unlocked.
I had no choice but to pick the lock. Unfortunately, my lockpicks broke, and all hell broke loose. I was arrested, and since I needed to escape the house, I let them arrest me. In jail, I slept my time off.
When I woke... I was outside Anvil. They shipped me across the ocean to go to jail for a measly 5 gold infringement.
Things get weirder from there, too, though I think it was a different mod at that point. A "Bounty Hunter" appeared and attacked the guard outside. For some reason. Lol.
So... Any idea what in the world THAT'S all about?

The doors lock and unlock
The doors lock and unlock because Morroblivion adds AI to the NPCS which causes them to do so in the evening and in the morning. You need to enter buildings during "business hours" which is usually between 8AM and 8PM. I don't know though why you were sent to Anvil, this certainly shouldn't happen. I can see from your other post that you use a lot of mods. Do you still use them? They might interfere with Morroblivion in some way. Especially if they alter crime.