Plumbing – How to achieve the correct garden tap orientation when threading it in

It's called a tap connector (obvious now I've found it), and you should be able to unscrew it. You can see one here

It's the bottom nut you need to be turning. It's part of a joint that attaches the pipe to the tap.

Once you have this off you should be able to remove the tap as normal.

If it's stuck then something like WD40 or penetrating oil might help. Failing that carefully heating the nut with a heat gun might expand the various parts enough to loosen them. Take precautions to avoid getting direct heat onto the bath (it appears to be acrylic).

A more drastic approach would be to take a hacksaw to the thread to separate it that way.

A shower drain needs a trap, so check under the cap on the left to see if there's a trap or a straight drain. If it's straight, then your assumptions are likely right that it's a toilet and the other one is a clean-out (fix that cap, you don't want sewer gases getting it). If you don't have a shower drain here, then I'm guessing it's only roughed in for a half bath.

I don't imagine the 2" would be anything other than a vanity drain.

With the "vent," I'm left wondering if it's really a drain since I see a clean-out installed on it. I'd recommend tracing that line up the walls, as best you can, to verify it really is a vent.

As for where the horizontal line is going to the left, perhaps a washer dryer, or perhaps the plumber screwed up the T and it should be pointing to the right for the sink. It's hard to tell from here.