Will I lose payments on a mining share in the time it takes to sync? I bought a mining share a couple months ago from Bitcoop, downloaded the wallet that they suggested, and the wallet has taken this long to sync. The first mining payout was a few weeks ago.

Will I get that payment or is it lost?
You will get that payment. Your wallet does not need to be online or synchronized to receive funds, only to do anything with them. (I'm assuming you're using Bitcoin-qt or something similar, I don't know what that vendor recommended or what you've done.)

However, the client shouldn't take months to sync, so this could indicate there is some problem.