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Since your Battlenet and Xbox accounts are already linked to your current Activision account, you can enjoy your progress even if you are using any platform that you wish for if you purchase the game again on your Xbox.
Yes, Call of Duty Modern Warfare 3 has cross-platform support. This means players can compete against and party up with players from any platform. These include PlayStation, Xbox as well as PC. With cross-platform support, it means you can play with friends who are not on the same system as you.