Strange things are happening here.
I disconnected the Vin pin. Connecting only the 5V pin from the 7805 stmicro voltage regultator I noticed that the board draw 0.45A! I jumped from the chair. I disconnected every output from it and it still draw 0.45A. It's hot as hell. I first noticed this problem smelling something in the air.. just a little smell of hot plastic... no smoke... touching the 7805 IC I understood something went wrong. I added a dissipator to it. Then I tried to touch the nano. It was super hot.
I disconnected the 5V pin, I connected the usb cable. Still super hot. I don't know if it has always been that hot since I bought it. The board is basically new so I don't know if it's a manufacturing problem.
When the usb is connected Vin = 4.4V
I bought a new one and in a few days it will arrive. This board will be returned.
It is not so strange... Remember what I said here below :
(Don't take this as an "I told you so", that is not my intend. I wrote this as info you will need when you get the new NANO.)
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(Question 1) With your 9v power supply, how many mA/Amp can it support?
The 4.5v you measured off USB supply is lower than expect, perhaps you are drawing more current than the 9v can put out.
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[bold added to quote]
Your USB has a current limit and your 9v power brick (combined with the voltage regulator on your NANO) doesn't.
So, when connected to your USB, when it exceeded current limit, the USB port would begin to lower the voltage to keep the current at 500mA which is the USB specified current limit. Whereas, when your 9v brick combination is overdrawn, it probably just poop-out to below what can support your NANO+plus your other stuff on the breadboard.
Now you use a 7805 which can push out a lot more than 500mA, things got cooked. Perhaps had your 9v been able to put out 1A+, the cooking would have begun with the 9v power supply,.
You have something not well there. It could be a simple short (one time I had a piece of loose wire got in the space between the breadboard and the bottom of the NANO), or a short/failed-component elsewhere on the NANO or the breadboard. The voltage regulators do fail - I had the voltage reg on a NANO failed before, which was found and replaced and the NANO is still in service.
When you do get your new NANO, first power it with your USB, that same USB port which worked (since it doesn't cook things) - solve that problem or your setup will just be a NANO execution device. Check the USB voltage before and after you plug in the NANO. If your USB is 5v+- 0.2v, and it drops all the way to 4.5v with the NANO again, you need to do more checking. If voltage is about the same as before the NANO is plugged in, that problem may be gone but just "may be" so you need to continue in a check-mode - now try the 9v brick again.
Only connect your 7805 (no current limit there) when you are sure that problem is gone.Oh, this is probably a lesson on the value of CC+CV power-supplies. CC is a misnomer there, it is really a current-limit instead of constant current. I would not hook up anything experimental until I set the current-limit to something I believe this device should not exceed (besides voltage, of course).
EDITed for wording error.