Only thing you can do is try to erase that HDD.

It sounds like an overheating issue. If there was enough dirt and debris it could have caused the fans to seize up when you blew it out. Do you feel any air flow or see the fans spinning?

IF you can wiggle the whole jack then that means that the internal structure of the port could be damaged. If thats the case, then the issue isn't with the connection on the outside, but with the solder joints connecting the 3.5mm jack to the board on the controller. You can try following the teardowns on the site and carefully inspecting the port once its apart. If you see any legs of the port are not connected then you can try resoldering it.

Hopefully its just the HDD. You can try replacing it to check. Just make sure to properly format it.

No, they are not interchangable.

If the controller turns on and stays on when connected to the console as well as works as normal, I have found that the Bluetooth chip on the controller can cause it to power off when it goes bad or gets shorted.

When you try to insert a disc, does it feel like there is a physical obstruction? Can you manually push it in with little resistance? PlayStation 4 Optical Drive Replacement - iFixit Repair Guide Physical Obstruction: Following the above guide will get you access to the drive. If there is any obstruction inside the drive like debris or another disc, you can manually free it by using the manual release. If it is dirt, then you can try CAREFULLY disassembling the drive (lots of gears and some springs). Any other: It could be a component on the drive board or the controller on the drive board. If that's the case, you will need a paired drive board/motherboard combo.

Usually with the flashing folder the HDD is on its way out. I would try replacing that and checking back in.

That pop sound is usually from a damaged or unseated cable. It could be any of the connectors on the board, but usually its the display. You can try opening it up to check the connections, but if those connections are good, then it could be damaged displays which can also cause popping. I would also double check that the unit can charge (Shows charging light) and that the connector on the board is solid.

When it powers off, do you hear any noise coming from the device? Like a popping sound from the speakers?