HAO DO I KNOW I GOT COIN>?

Okay, I'm being facetious, but I've tried to inform myself through research and now I'm just going to bite the bullet and ask some very newb-ish of questions:

Using Litecoin-QT, how do I know when I've successfully mined a block?
You get monies. Unless you've got some major power, you're probably not going to find one soon if you're solo mining

Also, I understand that transactions often are often accompanied by verification fees.  Since I'm running the client, does that mean I am participating in the verification process?  Will I eventually start accumulating a small amount of fee revenue as well?
no, transaction fees go to the person that mines the block

For bonus points:
I've read conflicting information as to whether the client needs to be open for me to receive coins.  It seems like I should be able to close it and when it catches up on the block chain, it will see I got paid.  So, which is it? Should I keep a client online if I'm expecting some coinage?
all transactions are in the blockchain. Blockchain is distributed. Close your client and others maintain the blockchain. Blockchain is downloaded off others when you reopen your client. So no, you don't need to keep it open.