These are symptoms of a greater issue. Wouldn't be wise to re-tile without first getting to the bottom of the problem.
Keeping what Bud said in mind as you demo, look for some of these specific errors, and take and post photos also:
Grout at the perimeter of the walls
Little thinset coverage under the tiles or tiles coming loose with there being little bond to either the tile or to the cement board
Cement board panels not being taped at the seams with mesh tape
No thinset under the cement board or construction adhesive used under the panels.
Once the subfloor is cleaned up, if the subfloor panels are not gapped, it's an install error or if you can slide a putty knife straight through, it means they aren't tongue and groove subfloor panels. You also didn't mention the age of the house, and if it's planking under the cement board, that would cause your problems also.
Let's help you get to the cause before you put one bit of new anything back onto that floor.