so, vanity- sand it. 100%. remove all the doors and drawers take them outside/garage and sand every part of it. Make sure you fill in all of the hardware knob holes w/a wood epoxy or good wood filler. it's ok if it's a little high because you're going to sand it flush w/the wood surface.
Now for sanding. I'd start w/a 120-150 and finish w/220, then a 320. super smooth. use your hand to feel the wood. wipe w/mineral spirits. get every bit of dust off. This is the time I would drill for new handle holes, so make sure you have those hand before you start.
Next, use your Zinsser oil base primer (spray can is much easier) and give an even, light coat. You want full coverage. so if a spot seems thin, give it a second coat. wiat to dry. take your sanding spongeof 220 and give the primer coat a fine sanding. use your hand to feel any rough spots. blow off dust and wipe it clean w/a damp rag. make sure there is no residue.
Now you can paint. get the best brush and lay down a nice coat. not too thick. let it dry. any rough spots of runs use a wet sand paper of 600 grit, (jsut run it under water) a wet sand (go easy) is the best way to get a smooth finish. when that's done apply your second coat. Make sure to use a good satin or semi gloss paint like BM Advance,
SW Pro Classic, INX Cabinet Coat, or, if you like, General Finishes Milk Paint also puts down a nice finish.
There are a few articles you can read on painting on this blog. She has step by step instructions (basically what I said, but more detail)
https://evolutionofstyleblog.com/?s=painting+oak+cabinets
https://evolutionofstyleblog.com/tips-tricks-for-painting-oak-cabinets/
Here's another tutorial for oak vanity. she uses the same primer/instructions I posted (just do NOT use the SW Emerald paint)
https://thediynuts.com/how-to-paint-bathroom-vanity-cabinets/
Shower door, I don't think you need a hole in the glass. it's not that big. just reach in and turn on the water.
I'm not sure I understand about your door swing. can it swing inside the shower?
Q