If they just removed the internal connections then they'd disable the USB ports but they'd also get notices of which staff were security risks / IT policy offenders (in the form of support requests "I plugged this USB drive in and ...").
Someone could hook up those ports with a new cable, but at that point they could also just open the case and bypass the glued port.
You can get PS/2 to USB adaptors which will at least allow you to use a USB keyboard. Probably not functionally the same as a USB port. Can't test this right now, but I wonder whether it would show up as a storage device if one plugged a USB drive into the adaptor?
No
In general those adapters are mechanical, and the keyboard needs a PS/2 mode.
No, the intelligence is in the keyboard, which detects the adaptor is being used and falls back to sending PS/2 signals.