Mining Bitcoin on Mars is totally infeasible, as almost all blocks mined on Mars will be orphaned by the time they reach Earth. I have talked about a solution before:
Each planet will have its own locally-mined cryptocurrency (Marscoin, etc), which each have a floating exchange rate to the original Earth-based Bitcoin. Marscoins and bitcoins can be directly traded for each other in order to settle interplanetary trade balances, the only issue is the speed-of-light delay for confirmations (which is unavoidable no matter what). Naturally Bitcoin can't be mined on Mars and Marscoin can't be mined on Earth, but that's not important.
Depends on Bitcoin hashing speed on Mars, if you have more Bitcoin hashing speed on Mars, Earth Bitcoin miners will not mine much
True, whichever planet has the largest hashing power will eventually dominate. Regardless of who "wins", it's not good for Bitcoin as interplanetary currency. Only if the communication delay is negligible compared to the average block-time, will it interplanetary mining be feasible. A coin with a 24hr block-time can be mined on Earth and Mars just fine (assuming a network connection can be established of course).