well for 1) the radar will be used to detect the aircraft get to know its direction speed altitude. then you launche missile facing the aircraft. so the missile is independent of the radar.
2)well you can use night vision.
1) The problem I was pointing to is that the speed of sound is way too slow in the atmosphere to use it as radar (like we use sonar <- sound in water)
lets do some simple calculations
The speed of sound at an altitude of 10km =>
speed of sound = sqrt( γ⋅R⋅T ) = sqrt (1.4 * 287 * 208) = ~289 m/s
So at about 30 second later the sound reaches you on the ground and the aircraft is already 9 km further.
In water at 10 Celsius:
speed of sound = sqrt (1.324 * 4186 * 278) = ~1 241 m/s
sound moves too slow in air, but moves quite fast in water (even much faster than what I wrote), thats why sonar is used (which is what you suggest, but in air), and not sound radars for air. However, there are probably situations imaginable where it could be used in combination, but the question is whether its worth the research/money/etc for what you get in return and whether something better doesnt already do a much better job.
2) Night vision works on the IR (infrared) spectrum, there are already quite a number of missile that do that. This is what they call a heat seeking missile.