Pretty odd behavior. Dunno if we'll be able to completely fix it, but let's take a stab at it.
Is it a big block Jasper? Does it have any internal memory? If so, remove the HDD and try connecting to Live with a profile newly created and stored on the NAND.
Do you have any friends/neighbors who would allow you to create a profile on their console and then try it on yours?
Before we look at your firmware status, we probably should check what's messing with Live.
However, I will mention that if original copies of CoD: Black Ops and Assassin's Creed: Brotherhood do not work, it's likely a drive issue.
What brand of drive was in there originally? Is it a refurbished console? (I wouldn't expect it to be since it's a Jasper, but it's good to know.)
What drive is in there now?