New vanity and moving the plumbing
The supply line and your drain lines are likely teed behind the wall, above the cut-out section. My guess is the vent pipe runs vertical out of the tee with the drain, so it won't need to move. You might be able to simply cut the pipes coming out of the tees, add couplers, and run new connections straight out. Or you might need to replace the tees with ones rotated properly. Before starting, make sure you know exactly what your plan is, ensure the water is shut off from the source, and if you're not confident in your ability to make straight cuts and new plumbing connections, call a plumber or a general handyman.

For your second question, it just means your vanity is expecting a single-hole faucet. These still accept both supply lines, of course; they just screw into the mixer valve under the countertop. If your vanity didn't come with a faucet, you'll need to buy a new one as your old three-hole faucet won't work (unless you can drill new holes in the vanity).