It this vanity mean to be temporary? If so, you could make it work depending it the bottom of the sink is high enough from the P/U trap, and far enough away from the bath tub. Not sure on current bath standards but it looks like you have 1" pipe and current standards (and sinks) may call for a larger pipe (what's called a tail pipe) diameter.

The drain pipe looks pretty fixed.

If the tile wall is coming down, you could cut the pipe and put in black plastic (ABS) plumbing to align the vanity but you don't have much to work with. I've even seen flexible pipe used though that's not recommended.

You might need a vanity without the lower shelf to give you more room.

I'd get a used vanity with sink/faucet and temporarily install it if you really need to have a vanity to use while the renos are being done. Otherwise just wait to replumb when the tile wall comes down. Those fixtures look pretty old.

Plumbing shouldn't cost too much but depends on your plumber/handyperson. Fairly easy to do if the walls are out. Easier if you don't have to move across a joist.

It is all possible but it depend on whether you want the pipe inside the wall or outside. Inside is a lot more work as you will have to cut through a stud (not recommended). Outside is of course unsightly.

How did a 31 inch vanity for centered on the plumbing? It must have been over a bit to the left? Are you replacing the tub? Id get a smaller one that doesnt bump out

Moving drain/vent lines on a gut job is usually not very expensive. Get the vanity you want, please.

The only piece you should ideally see for a single sink outside the wall is:

1) tail pipe coming down from the sink

2) U/P trap so one can clean out any blockages or retrieve dropped earrings, etc.

3) elbow and whatever pipe is required to go into the wall.

Shutoffs should just be poking out of the wall. Hot on left and cold on right of the the drain pipe. The lower shutoffs (w/o handles) should not be required.