Can sometimes take a while. Leave your bitcoin client open and be patient, it'll show eventually.
Correct me if I'm wrong, but if I'm understanding this correctly, I have to leave my bitcoin wallet client up and running in order for it to receive payouts/payments?
No, you shouldn't have to. Whenever you open your client it'll download the block chain and eventually your transactions will pop up. What it sounds like to me is that your coins from Slush are going to a different address (one that's not set up in your client). Did you have a different client prior to this one? Did you reformat your machine, or change machines and download a new client or something like that?