Users have done all of these things with eCart. I've done the first two before. I've done quickbooks export files, but "automatically" without import/export I'm not real sure about, but I think that is possible with some quickbooks implementations. It would be a matter of finding the quickbooks documentation on how that is done and following it. If quickbooks allows it, then it is possible with eCart (and custom php coding of course).
I could help with these things in a premier support ticket.