Drill a hole for a faucet in a porcelain sink

That's a drum trap, not a P-Trap. Be sure it's listed to comply with your code requirements. Looks slick anyway. You'll need a 45 degree chrome compression ell and maybe some extra pipe. You'll end up with the trap arm out of the drum running at a 45 degree horizontal angle to the wall. It will be cut quite short before it enters the ell. Upon exiting the ell, it should be a straight shot to the wall connection. The remaining trap arm you cut off may or may not be long enough for this. It then ties into the wall pipe with the same adapter fitting you would normally have used.

The trap arm runs entirely horizontal, despite the ell, with just enough slope to drain (1/4" per foot). The only significant vertical drop is in the tail piece that drops straight down from the sink. If you prefer the 45 degree segment to be closer to the wall, you could do a variation using two 45 degree ells, such that the trap arm out of the drum goes straight back for a distance before making a dog leg to the wall drain.

I would start by removing the trap, putting a bucket underneath the tailpiece (below the sink) and just running some water and seeing what happens. Clean out the trap while you're at it and make sure there isn't a mess of hair stuck in there. If it all comes through without a problem and doesn't back up, then I would suggest buying a snake and going through the wall pipe and seeing what you can pull out. I've always been able to get a slow drain going with a snake. It might take a few tries, but it can usually work out. Make sure that you push the snake right up against the pipe opening, and then just keep cranking until something changes.

You want to be careful with chemicals because they can discolor the chrome around the sink if you're not very careful, and they can also react with the pipes in a unsafe manner. They make some gels that are more organic in nature that can eat away at organic clogs (hair, etc) that are much safer to use.