Thank you for the detail. It helps a lot with the diagnosis.
When trying to upload the Rx LED blinks 3 times ...
That indicates that the driver is installed correctly
... but the Tx LED does not blink.
That can be either because the board type is not correct, or because the processor is not responding due to bad/missing bootloader or a defective chip.
Double-check the board-type selection once more to be sure. If you have the UNO selected and it still does not work, contact firstname.lastname@example.org
with a link to this thread.