if you ever have AS falling off the chip, then you have way, WAY to much on there. it takes a very small amount of as3 each time you apply the chip. do a search for how to apply thermal grease, cause you might be doing something wrong. it should never be just falling off the chip.
as for the as3 melting. yes, it does "melt." but this melting is usually a good thing. it takes many hours for most thermal paste to set properly. as it heats up, it will seap into the little tiny grooves in a heatsink and processor. this is what it is supposed to do. as it sets, it will not be taking up as much space between the 2 surfaces that you actually want touching. you dont want thermal paste covering both of the surfaces you are connecting. the metal to metal will have a much better thermal transfer. you want the thermal paste to fill in the little grooves where the 2 surfaces cannon meet.