Actually on second thought maybe changing the mass won't alter its orbit at all. And perhaps the only effect on the earth would be tides being stronger/weaker.

You could use a space elevator to get into space without burning fuel.

By the way, your first post's idea about angular momentum transfer between earth and moon won't apply. If it was rotated by rockets, their exhaust carries the balancing angular momentum. If it's done with a heavy rotating internal part then that's what takes it up.