You may remember that I recently discovered that there is no DME on the ILS approach to Rwy 9R at KMIA.  I was so flustered (it was one of my first night landings after a long flight) that I had to log on here to see whether it was my computer or FS.  I think we established that it was a bug in FS.

Well, I just tried to land at Rwy 7 at KJAX and there's no DME there either!

Is this some sort of recurring bug or does MS just hate Florida?

You might find this site useful:

http://www.airnav.com/

The relevant information for that particular ILS is here:

http://www.airnav.com/airport/JAX/ils/07

No mention of DME though.

The ILS is simply a beam pointing through a widening channel (as represented by the green chevrons on your GPS). If you're not in the area represented by the chevron, your radio can't pick up the ILS beam.

It doesn't know whether you're on the glideslope or not. Your Autopilot does by the angle at which the beam (which points upwards at the exact inclination of the glideslope) is received by the radio. It doesn't need to know how far away you are. It simply tries to keep the beam centred.

For example,  if the glidesloe was exactly 45 degrees (I know thats ridiculous - its just to demonstrate) and the aircraft was a mile away, the aircraft would be centred on the beam at 1 mile high. 2 miles out, the beam would be 2 miles above the ground and so on.