Can the same Steward be moved to each house as many times as you want,back and forth ?

I'm pretty sure that a steward is tied to the house you hired him/her at. But you could always hire another steward for each of your other houses.

No, and trying can really screw you up since (without mods or console commands) stewards cannot be dismissed except by death. Get different stewards for your other houses.

Trust me, don't even try it. I learned the hard way.