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).
Using bitcoins as interplanetary currency has been discussed here before, several times.
The only method that makes sense to me is for each planet to have their own version. Travelling from earth to Mars? Sign in to your local exchange before you go and trade some bitcoins for Marscoins. Mining bitcoins would only work for earthlings and mining Marscoins would only work for Martians.
This could get interesting when miners start hopping planets to take advantage of hashrate and price discrepencies.