Yes, it could be any of those things. You've got some sort of signal, so isn't likely to be a cable. The symptom you describe could be an overloaded input -- too much signal somewhere. If your amp truly has a line-level out, then it's likely you're overloading the input to your interface -- which you're almost certain to be doing if you're plugging a line level output into a mic level input on your interface. If it's one of the ones with a combination XLR/quarter-phone jack at the input, it may be that the XLR part is only capable of mic level and the quarter-phone plug is switchable between line-level and 'instrument' level, which is basically mic level but with a higher input impedance. Or some similar arrangement...you're looking for a line level input -- you will have to employ an adaptor to get from the XLR output to the quarter-phone input on your interface if it doesn't have some arrangement to switch the XLR input from mic level to line level.
If you just want to play or record with headphones, you could just plug directly into the instrument input on the interface and bypass the amp altogether, yes?