OK, so for receiving coins you don't have be connected to the internet, in fact you don't need to do anything as long as you have your wallet secured.
For both sending & receiving coins you need to connect internet.
For to just check the balance in a particular address you dont need to connect your wallet software to internet.
you can just check the address in blockchain.info for balance
But what about sending coins? - I assume internet connection is needed in order to broadcast the transaction, but do I need to download the full blockchain in order to successfully send coins?
Yes
Can I just open a fresh client for less than a minute, send some coins and close it immediately after that? Is that enough time for the transaction to appear in the blockchain?
No