HP LaserJet Professional P1606dn not recognized via USB on Windows 10

[Tech_guy_DR]

OS Build: Windows 10 21H1 19043.1526

Computer: Dell OptiPlex 7040

Processor: i5-6500

Situation:

I have installed Windows 10 to this PC since Windows 7 was the former OS. I've done many troubleshooting and this is basically my current situation:

• Under the Printer Server Properties the USB001 port is not listed nor any USB-like port.
• When I plug the printer in, the system won't even make a sound, display any device under the Devices and Printers nor update the Device Management list nor blink it.
• I know the printer should work because the user told me this printer was working but to start print it wouldn't start automatically instead the user had to press the printing button on the printer (this when it had Windows 7 installed).

For which:

• I've tried with many USB A to USB B cables I know they work.
• I've done the 'shutdown and with the printer turned on, plug the printer before Windows start' thing. It does nothing.
• I tried the printer under a different system (W10 19043.1415 HP Pro Desk i5-6500), the USB connection is completely unresponsive.
• I tried the 'create a '\\computer name\printer name' port and install the respective driver with Windows Update' thing and then the rebooting thing, really nothing worked at all.
• I tried to run the HP Setup in compatibility mode for Windows 7 and when the setup asks for the user to connect the USB cable it recognizes it not. 

Is there any way I can get the printer to be recognized via this USB port? I think the USB B port is just fine if it was ok for Windows 7.

[Tech_guy_DR]

Update:

I've tried to download the latest firmware by connecting the printer to the network. It did the upgrade from data code 20120814 to 20201023 (I didn't find any in-between version firmware to install it kind of incrementally, which I've preferred. If you have any other source rather than the this device's support page it would be nice to know).

Now the printer prints through TCP/IP but before it prompts for a 'Printer error' notification and then it prints. Additionally, the computer in which I installed the network printer is now behaving weirdly. The volume is going up and down and muting, also two File Explorers were present on task manager even with no File Explorer window open, the desktops icon started to blink and after killing them everything stopped except the volume error. 

I entered the device HTTP web page to see if any errors were displayed and I noticed the error on the figure below (Figure 1). Guessing to https://support.hp.com/us-en/document/ish_1689737-1117532-16#:~:text=10.00.-,10%20Supply%20memory%20.... there is some problem with the toner catridge, it looks legit but the memory chip is kind of damage I think (Figure 2). Do you people think this is such a problem? 

Figure 1:

Figure 2: Look closely to the small scratches: