They
go into the hand, which would trigger any cards which are allowed to be played as soon as you get them according to the rules and the instructions in the Critical Links and Useful Information sticky.
According to that info, Items can't be played while in combat (and by extension Running Away), even under the as soon as you get them rule. So no, a steed couldn't be played if you looted it.
If the other player had an
Elf card in their hand when they died, you could loot and play that for its run away bonus, though.