When I connect my Arduino nano to my pc, it is not recognized in the device manager. Originally, I thought this was a cable issue, however when I used a brand-new USB A to mini-B sync cable it still was not recognized. My Arduino uno rev 3 is instantly recognized with no issues so the USB port in my PC is operating normally. In addition to this, the nano is clearly turned on and the "L" LED is blinking like it should. From my understanding, when the nano is plugged into the PC it should immediately install its own drivers. This is not the case.
Yeah I recently checked to make sure all the connections were okay and everything looks good. The ATmega chip is does not feel warm and the FTDI chip also seems to be operating with a normal temperature. I only have it plugged into my PC with all of the additional pins open.
I've tried a new USB cable, a different port, and a different laptop. The only thing I have not tried is using a different nano because I only have one. I have used a different type of Arduino (uno rev 3) but that uses a different USB cable.
Ok, it's good you have tried those things. Most beginners don't think to try these things.
At this point, I am leaning towards @anon57585045 's diagnosis. Buy another Nano, from another vendor. At least then you will have a spare Nano, at the least.
Once you have a known working Nano, keep it safe. Don't use it in the the final version of any project. That way you always have something to compare against when things don't work as expected! I have kept my known/working Nano for almost 10 years now. Same goes for any non-trivial component. For example I wish I had kept my original DFplayer mini for the same reason.