I considered power might be a problem, but once the Native port is working, I can disconnect the Programming port, and the Native port continues to work, which tells me power is fine. I'm going to write a test program to see if I can figure out what it IS doing when only the Native port is connected.
As for the COM port numbers, I'm running Win8, which normally retains the same port number for a given device, regardless of which USB port it is connected to. I do sometimes see a device "Bossac Communications Port" appear as well. That one seems to appear when I try to program using the Programming port while the Native port is also connected to the PC.
I basically never use the programming port on the Due. Using the native port works perfectly fine for me. So, I'd still guess that something odd is happening with your PC or the Due itself.
As for port numbers - WIndows sucks. Even Windows 10 will give you different port numbers for different ports and it's annoying. It might be a bit better than Windows 7 where it is not uncommon for people to be up to COM40 or something. On my Win10 install I think most all my stuff registers at COM12 or less. This is as opposed to sane operating systems like Linux where devices are generally enumerated based on when you plugged them in, first come, first serve.
Also, the native port will register as two different com ports. One is the actual native port and the other is the BOSSA/SAM-BA port. You will see one or the other. If you erase the board it comes back up as BOSSA. If a firmware flash goes bad it might also come up as BOSSA or SAM-BA. Once the firmware is properly flashed it will be a different com port.