Shower – What to do about gap between tile and tub

I think your third idea will work and look the best.

First fill the void with expanding foam, allow the foam to cure, then cut it off flush with the face of the existing door trim.

Then cover the area with whatever molding/trim you find most aesthetically pleasing. You may wish to miter or cope the corners to get a more finished appearance.

Shower tiles are not a place to cut corners. If they are porous then you will have a host of issues. Basically you will breed mold faster and the tile install won't last as long - mold gets in cracks and pops tiles out or at least loosens bonding. You need to get a non-porous stone, plastic, glass, or porcelain.

If going with 12x12 tiles you will need at least 1/16th of an inch given they are uniform - and perfectly cut 12x12 are not cheap. So you are looking at 1/8-3/16. Which means you need sanded grout.

For cheap I suggest you go on craigslist or whatever is equivalent locally (even try job leftovers at tile store). The bottom 3 feet of your install needs to be a good tile - you will need about 34 sqft. You can get decent porcelain for $2 a sqft and even cheaper if it is a job leftover or end of stock. You can still use your ceramic up top. And when switching tiles I usually put in a narrow trim of glass mosaic in between - will need 3-4 sheets.