There is no need to sand down to the wood or to stain if all you are looking for is to make it black.
Rather than staining, you can just use a black paint. Issues with adhesion or intercom-stability between finishes are really non-issues. I would expect that the finish on the wood is polyurethane, although you might have lucked out and have an epoxy finish. Given the inevitable splashing and high humidity expected, I would suggest that you use either a polyurethane or epoxy product. Latex will adhere, but tends to build less and is prone to soften in high humidity. Given the small area to be painted, you are probably just as well to buy a spray can, rather than going to the trouble of diluting and filtering the paint for a spray gun. Given that you will likely be choosing a gloss product, and this will be highly visible, I would suggest that you will want to spray rather than brush it on.
If you are using a can, I suggest buying an external trigger. They are only a few bucks and make a big difference.
Lightly sand the surfaces to be painted and wipe the wood down with tack cloth. Mask off the glass (it is easy to scrape paint off of the glass, but you wouldn't want to contaminate your silicone). You will want to be in a well ventilated location. Apply the paint in a series of very light passes. You will want to hold the nozzle perpendicular to the surface, and move from side to side parallel to the surface (keep the nozzle at the same distance from the surface at all times, don't pivot with your elbow). Start your pass well to the side of the surface (12") then press the trigger as you approach the edge (6") and keep moving steadily. Apply lightly. If the surface starts to resemble an orange peel, you have laid the paint down too heavily - you are either spraying too close or moving too slowly - err on the side of fast and far to start. If you see runs, then you are applying way way too much.
After making a couple or three passes on one surface wait until the paint has just set (this depends on the open time of the product and a number of factors: humidity, temperature, etc. Check periodically on the overspray on your masking, and when it stops tacking (at all) wait a 10 or so minutes before applying the next coat). If you forget and wait too long, just give the surface a light sanding and wipe it down.
Repeat until you have the desired build. More thinner coats are better (keeping the surface flatter and smoother)