mine install always says 0 connections.
I see no option to generate coins, only sent and received tabs.
I'm wondering if a coin from Ebay will pop into the client automatically or do I need to have it check on the network.
Do you know?
No connection to the network means you won't get any information about transaction from the network.
I suggest both of you to read the wiki first:
https://en.bitcoin.it/wiki/Getting_startedIt actually answers all these questions.