The issue is not the "mobile home", it is the performance of the subfloor. Light weight subflooring needs to be stabilized to prevent flexing ... which leads to cracking of grout and tiles.
Others may have more experience, but for the floor I would use 1/2" Hardie Board and make sure you rigorously follow the screw pattern they require. Same goes with the gap and taping.
For the backsplash, and vanity, water proofing will be important too. Hardie Board with sealant (RedGuard or Dragon Shield) applied to the corners. You might want to call the Hardie people and get their input too.
Sounds like fun ... Good luck