I'm not sure what to suggest. Plugging or unplugging a USB cable should not shut down the PC and it is particularly odd that it should shut down using one set of ports but not others. That suggests to me a PC / USB port problem rather than a faulty USB cable, that it shuts down the PC even without the AXE114S connected suggests theAXE114S is not the cause of the problem.
I've always been advised not to run resistance and continuity checks on active components such as chips and USB cables as results can vary and damage can occur if not done using suitable equipment. The best way to test the USB cable is using the Programming Editor Serial Test then using another PC ( with a hub where a damaged cable is suspected ).
As to checking the download resistors, this can be done by measuring the resistance with the 28X removed from the circuit from the RX pin on the download socket to the Serial In pin on the 28X ( should be 22K ) and to 0V ( should be 10K ). Download can be tested using an RS232 cable where the USB connection is suspect, or with a known working USB cable on another PC known to work with that cable.
I know how difficult and frustrating it can be trying to determine where a problem is when it is not known what parts of a system are working or not. The best appraoch is to test each component of the system against something known to be working. That way what is not a problem can be eliminated. At the moment it is not clear what is at fault, the PC, the USB cable or the AXE114S, and what has been damaged by any failure which has occured.