Is it possible for a player to be eliminated after losing his home system, all his units and his last planet? Do they kep selecting strategy cards and get VP's from the Imperial strategy card? Or do they just leave the table and take no further participation in the game?

Elimination is possible.

From the FAQ:

Elimination:

A player who controls no planets and has no units on the board is eliminated from the game. He cannot play Action Cards or choose Strategy Cards. After a play has been eliminated, the other players continue the game normally, with no changes to the number of Strategy Cards taken, etc. When a player is eliminated, all his Action Cards are discarded and all his race's Trade Agreements are removed from the game.

It is worth noting that essentially you stop participating in the game, but you still exist.

You still have your victory points, and if the game were to end, your score is still factored in.

Also, for objectives that deal with your left or right neighbor, the Eliminated player is still considered your neighbor.

He has 0 influence and his homeworld still functions the same way.