DI is direct in. Bass straight into the mixer without an amp. I think you answered my question anyway. If it sounds good through the phones, then the problem isn't the bass, cable or mixer, right? The problem probably isn't CEP either - it's pretty reliable most of the time.
Pops and clicks seem to be an inevitable part of hard-disk recording right now. I get them regularly, but not frequently. That's why Cool Edit has a click/pop remover. But if you're getting them every couple of seconds, then there's some major problem - my guess is something to do with the computer, but I'm no whizz at this stuff.
Suggestion: post this problem in the Computer Recording Forum, where lotsa experienced people hang out. I'll follow the thread - I'm interested too.