Once a sender sends you a transaction, how long generally until it shows up in your wallet, assuming it has not been expedited?
It can show up pretty quickly sometimes but can take a hour or two at other times.
'When' it is sent is another thing, when you are talking about those free bitcoin sites.
Some seem to send you your free coin right away while others sites may wait to send
many payouts at once to save on transaction fees... some won't show up for a day or
two.
I have tried some of these sites but have not seen any coin yet, then again my wallet is still 'updating' so is that the reason possibly is my transaction has not hit it yet?
Yes you need to wait until you have caught up before you see some of your most recent transactions in your client.
But while you wait you can use
https://blockchain.info and do a search on the address you used in order to see if
a payment has been sent to it yet, and how many confirmations it has etc.