Why are AMRAAMs often mounted on the F-16's wingtip hardpoints?
The AIM-120 weighs in at about 335lb (152kg), and the AIM-9 at about 190lb (85kg). While heavier, the Stores Management System (SMS) and the Fire Control Computer (FCC) know what is installed where, and compensate/adjust flight characteristics as needed.
You put the most capable missile, the AIM-120, out the outboard stations. You'd use that first. Only later might you want to use the shorter range AIM-9.
So, if you were to shoot the AIM-120 from Sta 2 (left inboard), you'd still have the AIM-9 out on the wingtip.
Do it the other way around.
Also, the AIM-9 since the AIM-9G can slave the missile seeker head to the aircraft radar. Target acquisition is not solely dependent on the IR in the missile and line of sight.

The AIM-9G was the first version to include SEAM (Sidewinder Expanded Acquisition Mode), basically an improved AIM-9D seeker head, but was quickly made obsolete by the AIM-9H. The latter was the first model to offer limited all-weather capability, as well as solid-state electronics and double-delta control surfaces resulting in a superior maneuverability compared to older models. SEAM slaves the seeker head of the missile to the radar when in &quot;dogfight&quot; mode, which enables the seeker head to be uncaged, slewed toward a specific target by the aircraft radar, and made to track that particular target before being launched.

(from F-16.net - http://www.f-16.net/f-16_armament_article1.html)