my spare sits firmly on my bonnet, not got one yet, but will also be carrying a spare inner tube when I get round to it.
as to tyre pressures, monthly before I go greenlaning, fluids weekly (just a habit I have carried on since I started learning to drive & had some right sheds of cars the NEEDED topping up weekly)
in the long term, no it won't do your diff or transfer box any favours running different sizes, but as a get me home its not going to make much, if any difference.
Just as a space saver is supposed to get you to a safe place to get your tyre repaired, not to be running on it for weeks on end.
My wife hit a pothole and ripped out the sidewall of the tyre thus a can of gunk would've been useless.She rang me from just down the road,off I went to rescue her. Had the wheel changed and her on her way to work in less than 20 minutes. The car only has a spacesaver spare so she had a new tyre fitted in her lunch break.Job done.
I spent ages with my ancient foot pump getting my tyres to the proper pressure for towing only to notice a bulge in the side wall of one. Off I went to my local tyre fitter who popped a new one. When I got home and checked he receipt,it noted that the fitter had let all my tyres down to the "correct" pressure!
------------- Silence is golden
Duct tape is silver