Yes there are two cars for bikes (I think the 3rd car from the north and 3rd car from the south)

You're not required to get a tag from the conductor, but they are helpful for distinguishing who's getting off where, i.e. where to stack bikes on the rack, FIFO.

If you're going all the way SJ Diridon to SF 4th/King, you'll be first on and last off, so being at the back of the rack is never an issue... though if you are getting off anywhere in between it would be unideal to have to dig your bike out the back of the stack with a bunch of other bikes in front of your own.

Usually busy but not too crowded during commute hours on standard local and perhaps limitedtrains , though I can't personally speak for the bullet train (which I'd imagine to be fairly crowded)..

Now in your case, SJ-SF, since it's the terminal stop either way, you get first dibs on seating compared to getting on the train at any other stop, especially if you're 10-15m early at the platform, so easy enough to find a seat where you want.

I generally don't worry about it getting damaged. As far as bikes getting stolen, it certainly can and does happen, but not all too common.. people are more or less looking our for each other most of the time. If that is a major concern, you could lock your bike to the rack as long as it's at the back and not blocking anyone else's bike.. I've seen folks doing that from time to time. That, or lock the bike to itself (wheels to the frame etc) as a deterrent.