The tile in my shower is the same tile as was installed all other areas of my house. The tiles are 13"x13" and the grout seams are approximately 1/2" wide. The grout is a rough, sandy-type texture. After the tile installation and prior to use, grout sealer was applied on three separate occassions. The shower grout never completely drys between use.

The shower floor is not level from corner to corner as the floor dips to a gap of 1" or more at the each of the four wall's center points between the base of the wall tile and the floor tile.

The contractor used white silicone caulk to fill the space between the wall tiles and the floor tiles leaving an unusually large caulk seam (1" or more wide) at the four midpoints of the walls. Within only a few weeks of use, mold began forming within the caulk and was not accessible form the caulk's exterior nor removeable with any househould cleaner.

The shower floor does puddle water which eventually drains or evaporates.

I have removed all caulk between the floor and wall tiles.

Should I replace all grout within the shower floor and if so with what should I replace it?
And with what should I fill the space between the walls and floors (grout or caulk)?

To be honest with you I'd rip the whole floor out and do a better job of installing. Grout should never be 1/2", 1/4" is the norm for 12 or 13" tiles. Also, when tiles end close to a wall like yours does the first tile should be shifted half tile to either side so you don't end up with cracks 1/2" wide. I would also recommend a 2 part epoxy grout that cures like stone and never holds a stain.