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They'll miss out on Europe even if Villa don't win and Arsenal stay in the top 4, as the cup runner up get the Europa spot if the winner finishes in the Champs league spots.
No, UEFA abandoned that this year.
Only the cup winning teams get a place, if they've already got a place (by finishing in the top 5, now top 6 since Chelsea are guaranteed a Champs League place) then the place goes to 7th in the Prem.
[EDIT] Or, more simply: http://www.bbc.co.uk/sport/0/football/32358581
Villa to win the FA Cup and Libpool to finish 7th in the Prem and miss out on Europe?
Or, as in the order it would happen; Libpool to finish 7th in the Prem and miss out on Europe if Villa win the FA Cup.