Lay the tiles and cover them. You will not damage the tiles if you cover them with cardboard but it would be very easy to damage or stain your paint with grout or the edge of a trowel or a tile.
It doesnt matter at all. Some builders do paint first some do the tile first. As for staining paint... Ive never had that happen to me in all 16 years but I do wipe it off fast.
Ah, but you're a pro TileGuy
I wouldn't worry about my walls if I had you laying my tile, but for the average homeowner, it would be easy to make a mistake.
In my opinion, it can be done either way if you are installing base trim. Personally I would texture the ceiling first, then install flooring, paint walls and then install base trim. Thats just the usual order I was taught to work in.