The same as any other PVC drain. Trailers have a plastic or fabric covering underneath. Cut this at the drain and repair the pipe. Duct Tape the cut in the fabric or staple a section of cheap blue tarp over that section.
It will connect to the PVC pipe into the wall which ultimately will connect into the main sewer line outside.
Bathroom sinks use 1 1/4 pipe for the downspout in the sink. You can use 1 1/2 from that by using an adapter washer in the connection above the trap.
Vent, return circulation
It can if the plugged part is actually down the pipe from both, not actually under the sink. It sounds as if the main line connecting both is plugged. See if you can find a 'cleanout' on the main pipe and 'snake' it.
If you are having issues with leaking under your sink, it could be as simple as a loose pipe fitting. Try first to tighten all of the screws. If you still experience leaking, you should contact a local plumber.
A leak
Depends on how much is flowing into the pipe. One bathroom sink, 1 1/4"
Yes a clogged VENT pipe can cause a sluggish drain
You can extend the faucet pipe by, getting an extra metal pipe and welding it with the faucet pipe and it shall be extended.
You need a pipe wrench. Try to borrow-they are pricey.
There should be a nut around the chrome pipe holding it to the brass. Loosen the nut and pull the chrome pipe out.
A poor plumbing job. The pipes to the shower & sink share a pipe to the water heater and the pipe is not large enough. This causes the water pressure in the pipe to drop when the hot water is turned on at the sink while the shower is also on.