Lately, my xbox has been kicking me off of Xbox live.

I don't believe there is anything wrong with my connection, as my games have zero lag, and as long as I'm playing a multiplayer game then there isn't an issue.

My Xbox disconnects after I leave it idle for a few minutes, or while I play a single player game. This is frustrating, as I'm currently trying to re-download Borderlands onto my new hard drive, but I have to keep manually re-connecting to Xbox Live to continue the download.

I don't understand; my connection is perfectly fine when I play a multiplayer game, but as soon as I put in something single player or idle to download something, I get kicked off.

Does anyone else have this problem?

Try resetting your router to see if that helps.