Under what circumstances would an outlet start to melt - not burn, melt - without throwing the circuit breaker?

Other than the obvious - it's a faulty breaker (which it is new).

Is it possible that a minor short below the threshold of 20 amps can take place which is hot enough to melt the plastic covering on the outlet?

Asolutly !
Theres only one thang that can make it melt and that is a slow, slow short, that is not big enough to kick the bracker... Letting it go on can cause some serious problems....
Solution?
Replace it....

Assuming that the outlet in question is a convenience receptacle, and that is what's melting, the obvious first step is to replace it - as fritzz says. Any ground fault, again assuming it is not a GFI receptacle or on a GFI circuit breaker should trip the breaker even on the tiniest current, in the range of milliamps. So the most likely thing I can think of would be a high resistance either in the connections, terminals, or the actual contacts between the receptacle and whatever is plugged into it. If the prongs have lost their spring or the plug prongs are bent or corroded. A 20 Amp branch circuit can carry anything up to 20 amps which represents up to over 2KW of heat - more than enough to start a house fire!

That answersmy question nicely. Thanx