This threw me a bit, since a side panel isn't a door.
However, assuming it was a typo, it doesn't really change much of my shower tray installation advice. It depends on whether you want the tray to have a small step in or not. The only complication is the waste pipe, the exit through the walls being as previously described or by drilling a hole through the centre of the joist and the the wall (you'll need to hire a core drill for this) and attaching a new boss strap to the soil pipe and sealing up the existing connection into the stack. Is the stack plastic or cast iron?
The other way, which utilises the existing waste pipe will require you building a raised platform (from 3 or 4 by 2 timbers), which you can screw directly to the joists at right angles to them. This will then be covered by ply and the tray will be bedded either on mortar ar directly onto the ply (read the instructions that come with the tray - they usually specify their preferred method). If you choose this way, then the frame itself (and thus the flooring ply on top of it) can be made precisely level by packing wedges between the timbers and the studs as you build it. The trouble with this frame will be that you'll have difficulties connecting the shower waste to it, as you will have to create an elongated "U" trap from plastic pipe which has to go beneath the timbers of the frame and up to connect to the existing pipe.
If it were me, in my own house, I'd bite the bullet and go for having the tray flat on the floor - you've gone to a lot of effort so far, and a bit more is neither here nor there. (Personally, I'd replace the stack if it's cast iron, and use new soil pipe for the WC since removing it now will make it much easier to drop the flooring down and under that stud wall near the door. But then, I would wouldn't I, but I don't mind heights too much)
With regard to the wayward studwork - there is nothing to stop you putting some (drywall) adhesive onto the adjacent studs, putting plasterboard up and screwing it to the stud that stands proud and adjusting it so you create a perfect right angle in the corner. Leave it a few hours to go off, then screw the board through the adhesive into the end studs. Thus the wall will be true and square, which is of course highly desirable.
Final thought for the time being, especially with regard to the window - you should really have building reg approval for this work (who will require details and calculations for what lintel has been installed to support the wall and roof above it). I know it's a refurb of an existing room, but the wiring (should be removed) worries me a bit- this, the cross bonding, the replacement of any cables for an upgraded electric shower, and the installation of an extractor fan really should be done by an electrician, which won't cost that much really in the grand plan of things.
