Remove all the contents and drawers from the vanity.
Turn off/disconnect the plumbing.
Look inside for any screws that would go into the walls and remove those. Specifically around the top, base, and sides of the vanity. You might find how the sink top is attached to the base as well, splitting those will make it lighter but maybe not easier. 100% depends on the assembly.
Cut the caulk between the wall and vanity top. Probably goes all the way around and maybe under the lip.
Lift up on the corner of the top/vanity. Everything should lift up and start breaking away from the wall. It should lift with some weight but you shouldn't have to really leverage on it to move it. If it still feels attached, look for more screws.
To add to step 4: after you cut the caulk try and slide a putty knife between the counter top and the wall to check for adhesive on the backside. If there is adhesive you can try and cut through it with the putty knife that way you don’t rip/peel the drywall as you’re pulling the counter out. The less damage to the dry wall the better.
As for the texturing it is called knock down texturing. It’s done with dry wall compound. Here is a YouTube video showing the patching steps video
Check YouTube for videos of vanity removal they will walk through all the steps above as well.
May have been glued to the wall as well. I would expect to have to repair the drywall. For the texture, they have spray on texture in a can that they say replicates the existing look. I would not expect to get it perfect. It will probably look different no matter how hard you try.
It's probably just construction adhesive holding that part to the wall. So you can try prying that off first but sometimes they attach it with something else to the top (besides just caulk).
No it's likely glued to wall. It can and will rip the drywall. If you get a large thin scraper you can try to push it between the wall and splash guard. (Like a 12" taping knife)
And matching the texture will be very hard. It looks like some type of knockdown or medium orange peel. It's possible the texture is in drywall and was done before vanity installed.
If you're up for a lot more work, you can skim coat the drywall to remove the texture. It's takes a good bit of time and you'll need to sand so there's a lot of dust. I doubt it's the way you go unless you don't like the texture, but it's an option.