I never had alternator problems on my 2005 GT, but several posters on this site have said they needed to replace the alternator on their Mustangs. It sounds like yours may be getting weak, but its hard to tell how long it will last.
With the engine running, HVAC blower on high and headlights on, it should be able to maintain approximately 13.8 volts at almost any speed above idle (it probably puts out full output at about 1500 rpm and up, but I would have to check the specs). If you can get that 13.8 volts out when running, then it might just be a weak battery.
Most of the auto store battery and alternator testers only make a very gross-level measurement of good/bad. I had an alternator in my GMC truck that would only put out about 10.5-11.0 volts if the headlights and HVAC were on, but Autozone still showed it as good because they never tested it with a load. I finally put a separate voltmeter gauge on it and watched it during normal driving. I was driving to work in the dark in the winter (lights and HVAC on), so the battery never really got a full charge in normal use. Replacing the alternator solved all my charging and starting problems.