Does the quality and rare rate when an item drops change due to whom gets the final blow?
No, the drop rate is not dependent on who actually lands the final blow.
Diablo 3 does not use the same magic find as Diablo 2 did. They used to use all kinds of things like magic find % on items, increased magic find for solo players...etc. Now, they use a group-magic find increase - if you are close enough to other players the more your magic find is increased. All magic find is shared and is a % based on your difficulty that you are currently playing + your group magic find bonus.
All magic find on items are gone. It is impossible for someone else in your game to have a higher % magic find unless you are too far away from everyone else to lose the group-magic find bonus. You can see when it is enabled by an icon on your hotbar.