Man. I'd get a book on bathroom remodelling and go from there. I have a house built in 1922 and it is in a CONSTANT state of remodelling. We haven't made it to the bathrooms yet but we will. You'd be surprised what a good book will contain. It will most likely address the very situations you are in.
Anytime I start a project I go out and look for books first. A/C; electrical (shh); putting in a load-bearing beam. There are books for everything.
The gap between the tile and the cabinet might be disguised by a piece of molding. I'm not sure exactly what you are dealing with.
Anyway; as far as re-tiling the floor. I'd remove the toilet. Yes it sucks, yes it's nasty but you probably need to do it anyway. You'll have to make sure that you can replace it. They make an adapter if your flange is funky.
The rotting floor around the tub is another matter. Are the joists rotting or just the subfloor?
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