How close for the toilet and vanity

You must have a vent. The toilet will not flush properly without one, and there will rarely be enough water left in the bowl. You need a 2" vent to take off from the branch line within 6 feet of the toilet. It can run up any convenient wall until it is above the toilet's flood level. Then it can run near horizontally (1/4" per foot) anywhere to tie into an equal or larger vent.

There's any number of ways to properly vent a toilet, and several common ways that are technically wrong, and even more that are just plain wrong. It's impossible to suggest a good method without a lot more information.

Cut that pipe out in 2 places. One place is above the bottom shelf of the new cabinet and mid way between the 2 fittings on the upper pipe that goes at a 45 degree angle. Recouple it back together when the cabinet is installed. The supply lines need to be relocated too, but you may be able to move them around to get past their issues.

The cutting of the vent pipe is still taking a small chance it will be difficult to get the pipe back together. As it is you will need to shorten the pipe you cut out at least by a 1/2" to make room for the stop built in as part of the coupler. You will also need to account for how much out of square the cut has too.

If the upper portion of the pipe has a little upward give to it, it will be a lot easier to rejoin. If not you will need to be really assertive on getting it back together again.