Broken/unprogrammed UNO R3? =(

Since the Device Manager is showing "atmega16u2", your Arduino is in DFU mode.
This indicates the atmega16u2 does not have the Serial to USB code loaded on it.

If you want to try to recover it yourself you can use the following page as a guide.

The instructions are the same for the 16u2.
Since you are on Windows, you can download and install FLIP. (The link is on the page above)

Download the following firmware and install this on the 16u2 with FLIP:
https://github.com/arduino/Arduino/blob/master/hardware/arduino/firmwares/atmegaxxu2/arduino-usbserial/Arduino-usbserial-atmega16u2-Uno-Rev3.hex

After that, you should be able to install the UNO USB serial drivers.

Hopefully, the bootloader is installed on the atmega328.