If you search for bathroom wall cabinet, you will find lots of different kinds. Restoration hardware has a really nice one - but similar ones can be found for a lot less. If I find one I like with glass doors, I plan on frosting them.

yes it is meant to recess between existing studs, is skinnier that a stud space so there is wiggle room. it sticks out @ 1”. Here is a pic of the Signature Hardware one we did. I hate to complain because it looked fine, fit the space well, easy and relatively cheap. This was for a last minute quick update of a bathroom before listing in 1 day, not committing to any serious work. It’s skinny (about a foot) and easy enough to cut a hole in drywall & pop it in. top pic is my phone pics B&A the 1-day makeover, 2nd pic was from the realtor pics. i doubt the paint on that cabinet was even dry. ETA yes we put the hinge on the left so the door did not hit the light. That part was fine, what I didn’t like was the body of the cabinet was a flimsy plastic shell. Plain stud space would have looked better inside.

Several ideas: Idea 1: A shelf above the bathroom door could provide storage for toilet paper, and baskets can "neaten things up". This would only be visible from inside the bathroom. Idea 2: You say you're against too much clutter above the toilet. How about a train rack for towels? You can get a small, neat one, and the linens could bring some color into a rather empty bathroom. Moving linens to this spot would free up space under the sink for the toilet paper. Idea 3: How about a basket near the toilet to hold toilet paper rolls? It brings in an organic, natural item that softens the all-white bathroom.

Figure out what really needs to be stored in the bathroom and what can be moved to other areas of the house. The sink is not pictured but if the sink is a cabinet consider moving most storage under there and install pull out tiered shelving in cabinet to maximize space and make retrieval easier. A hand towel over the toilet gives me the heebie-jeebies yucky ! As noted in photos below move the light to above the mirror and have mirror same width as sink to feel more expansive. You could also do the whole space with a mirrored cabinet vs a mirror and cabinet so its one cohesive expanse doing a mirror and a cabinet may be choppy