Well, no team has ever won the European Cup one year and the Cup-Winners or UEFA Cup the next. It couldn't be done, the winner always gets into the next year's competition.
And no team won the Cup-Winners' Cup one year and the Fairs/UEFA Cup the next.
They've recently changed the rule to allow teams to get into both if beaten in the CL, wrongly in my opinion, so I guess that nice Mr Abramovich could be the first owner to get his hands on both trophies.
If they do win it, they'll hold both trophies for 10 days - the Europa final is 15th May, the CL one is on the 25th.