That is a strange problem indeed. If there are no connections to the printer and it is still firing off a job at boot then there is a glitch somewhere on the printers memory. Something is forcing the printer to think it needs to print a job when none actually exist. Adding to the confusion is how the printer complains of a jam in tray 2 but is somehow impacted by the paper loaded in tray 1.
If you have warranty time left on this printer then now might be a good time to call HP for support. They should be able to work with you to reproduce the problem and let you know your replacement options.
If you are out of warranty then the best you can do is try to address the paper jam situation on tray 2. The printer may think that a job is jammed in the tray 2 paper path. When the jam is "cleared" it then runs a page through the paper path to make sure it is actually clean. Id take a closer look at the paper path around tray 2 to be safe. Check out all the sensors and rollers in the area. Swap the tray if you happen to have a spare lying around from another CM2320 and see if it reproduces the problem or not.
Im doubtful a firmware update would have any impact on the situation but if you happen to be out of date then you could try upgrading.