An all weather outdoor shower is easily done by locating the mixing valve inside the house and use a coupling shaft to locate the handle on the outside. You must use a standard tub type lower faucet (with the pull up to activate the shower) to allow the system to drain out when the shower is done. You also have to make sure everything slopes in direction of the water flow. We live in Bancroft Ontario so winters are well below freezing and we have been showering outside for 3 winters. Only issue we have is the shower head will freeze so we typically remove it after a shower or shower without it on when really cold. The system has worked well and it helps that nothing special needs to be done by anyone. no inside valves to close etc. Just turn off tap and everything drains.

~$400 including expedited shipping. My GC just had his regular countertop guy do them, which wasn't great.... he had trouble making clean cuts. If I were going to do it again I would have pressed for someone with expertise to cut them. You can see in the image that the facing is not made of the stone peak, because too much was broken during the cutting to make the mitered edge. I'm a little bummed, but mostly pleased with the surface (which is the important part) thus far.

Don't use the wall faucet variety, UNLESS you want to purchase a new top for the vanity. Or return the vanity. Pretty simple. The round hole was meant to have a faucet. Not a "cover" which may also allow water down under, which may also ruin the vanity as there will be no containing water lines beneath the little hole. Bad idea,

I agree you probably need to centre the vanity and the cabinet on the wall or they could look a bit odd! You don't say why you don't want a space between the window and the vanity - just my opinion, but I would think 300mm could be quite useful for a bin, a clothes basket or box, or a small stool for putting clothes on? The only alternative to leaving a space, is to fill it with something as the response above suggested. To avoid it interfering with the window, you could make this something lower, so it sits level with the top of the vanity. This could be built in and the wall lining put over it to create a single "shelf" level with the underside of the vanity top, or if you can find some freestanding open shelving of the correct width, which could also be cut to the correct height, you could wedge that between the wall and the vanity side when fitting the vanity. Just bear in mind also, that you'll lose some of your towel hanging space if you put something too high in the space.