If the mortar base is flaking and chipping, then yes it needs to come out. It may be better to remove the entire mortar base and put down cement board for the new tile. Otherwise, patch the areas where the screen is showing.
I agree with Bud about fitting a door to the opening. As for the windows, you can get the size(s) you need made to fit the opening (vinyl). You can even have them mulled together and install as one unit, no need for a center post.
Your method for locating the squeaks is correct. The noise is caused when the plywood moves up and down on the nail holding it, which has become loose over time. I would drill a pocket hole in the joist and screw up into the plywood. Maybe even 2 screws (one on either side of the joist)...
Yup, draw the profile on the door (copy location from your old door).
Then go over the outline with a utility knife and next set the router to the depth and freehand the cutout. Take your time and keep the router flat to the door edge surface (no tipping)
If you remove the joists, do so after getting the jacks in place. The gable end wall wouldn't be a problem but the sidewall might if you try and jack it after the joists are out. You could take a measurement from a fixed beam and check it every so often to make sure the wall doesn't move...
Yes, that is a good plan. I only do one wall at a time when doing this type of work. You probably will find the new lumber is narrower than what was used so it should fit fine and may even require some shimming.
Jack as Bud shows and replace a piece at a time (5' / 6' or so). If there are multiple thicknesses, make sure you stagger the joints at least 1 foot. Of course do one wall at a time. And put some vapor poly down over that dirt before you cover it up.