What is the rough estimate of internet speed needed to run the standard client? Does a faster Internet connection increase the client performance (by that I mean how fast it updates its blockchain, sends transactions, and how well can it function as a node in the network)?
Internet connection speed has to be fast enough to download new blocks as they get mined. With current transaction rate it requires 1-2 Kb/s. But faster connection would help to reduce chance of double spend attack if you run retail business and accept transactions with 0 confirmations.
The speed at which you download a new copy of the blockchain is more dependent on the # of connections you have with the network. To increase the number of connections to the Bitcoin network and improve Block chain speed open/forward port 8333 to the computer which is running the client.
This will reduce the delay (latency) between the time new transaction/blocks occur and when you learn of them. Having more connections means on average you are closer to any event (the a node submitting a new transaction, or new block solution) and thus learn about it quicker.