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Ad the title says... How do you interpret the interaction between psychic power death mission and remnant of glory phoenix gem?

They seem to interact just fine, Death Mission removes the psyker as a casualty, and the Phoenix Gem triggers when the model is removed as a casualty.

What does happen when if the farseer comes back to life?

Well death mission is over so why not come back from the gem, however I would interpret the rule when the game is over its over no coming back.

Suppose he comes back not at the end of the game, can he cast different powers?

If death mission has ended, then yes.