New guy here - Former F-15C pilot with a few answers.
Without question, the missile is an AIM-7M. It’s likely there is a second AIM-7 on the aft fuselage station out of view of the camera.
While Harms may have been tested in the early days of the aircraft, they have never been carried by operational F-15Es. The aircraft is not equipped with the interface required to use the missile.
There was a period of time during the initial years of the transition to the AIM-120, that they weren't plentiful in certain theaters. I'm not sure about 1999 but it's possible this particular unit was allotted only so many AMRAAMs(due to their primary mission as a bomber) and rather than go out with empty stations, they chose to take some AIM-7s as well. Personally, I'm surprised to see them using one of the fuselage stations for an AIM-7 as that weapons station can be used for bombs or missiles. I would think, in light of the limited number of aircraft patrolling the NFZ at that time, having more air to ground ordnance available would have been a priority.
There are several reasons that a mix of AIM-7s and AMRAAMs was preferable (more so in the F-15C). Some can be discussed here, others cannot. First, the AIM-7 is a much larger weapon with a bigger warhead. If I had to down a larger aircraft, I might choose it over the smaller AIM-120. Also, due to its slow acceleration compared with the AMRAAM, the sparrow was surprising capable in certain types of turning engagements. When stuck in lag with your nose well behind the tail of an adversary but with a radar lock, the AIM-7 was the weapon of choice. Another was mentioned in another post concerning friendlies nearby, although that situation isn’t as likely as it sounds. If you can't shoot an AMRAAM because friendly aircraft might be in danger from your shot, then it's unlikely an AIM-7 would be much better. In reality, the scenario described where friendlies might be nearby and a factor to your shot was very limited in scope. Both missiles are very discriminating and if shot with a solid radar lock, the chances of them hitting something you don't want them to hit are very small.
We would never choose an AIM-7 over an AMRAAM for "economic" reasons (i.e. let's get rid of them because they're old). We didn't train that way and we certainly wouldn't employ that way either. Frankly, the chance to engage an enemy fighter is so uncommon that a fighter pilot is simply going to choose the best weapon for the situation and press the attack to the utmost of his ability. The shelf life of the weapons he has on board will probably never even enter into his thought process.
Any version of the F-15 can fire wing mounted missiles while carrying wing tanks without restriction. There was a restriction to firing fuselage mounted AIM-120s and AIM-7s with wing tanks still on. The aft mounted missiles might contact the tank under certain low G scenarios, so the solution was to jettison prior to launch (likely due to normal tactics approaching an engagement) or load the jet up with a couple of Gs during the shot.