Who told you that..it can be fired even in close range..not only AIM 120..any missile..Iranians have confirmed kills with Phoenix at few KM only...if I remeber corectly AIM 120 has 2 or 3 km minimal range..Phenix Aim-54 is also around 3 km...These active radar guided missiles have active radar seeker and these missiles are fire and forget only in this range,everything that exceed missile onboard radar has to be compesated with mid course guidance.
But at same time these missiles can use own radar immedieatly after lunch and this black spot of 2--3km after lunch is only limitation, even it makes no sense to use it if you have better performed and cheaper short range missile,those kills I mentioned with phenix were forced after AIM-9 failure or simple out of IR missiles
I'm not saying this Indian claim is true,this is just corection regarding your claim about aim-120 since there are lot of incorect information floating online...any way,when it comes to Indian claim, aircrafts don't vanish after been downed,if F-16 is downed it would show up some where....it is hard today to hide loss of aircraft.