What does Chess.com do in this situation?

I've had knight vs knight in a bullet game quite a few times and it doesn't automatically declare the game drawn until one player's time has expired.
So I think in the given situations you would undoubtedly get the mate as it would definitely allow the final move to be made.
The problem that you may have is that if the opponent realised what was happening is they could just wait for the time to expire rather than making the move that would allow mate - that would enable them to get them draw.

Even if time expires for one person though it should still be a draw because in order to win on time you have to have sufficent material to mate.

If you don't capture the opponent has still sufficient material. If you capture the opponent has a mate in 1.
So the problem doesn't exist.

Even if time expires for one person though it should still be a draw because in order to win on time you have to have sufficent material to mate.
I think the problems proved there is sufficient material to mate, because with only that material, there was checkmate. Pretty self-explanatory.
I'm still confused. It's possible to mate (although not to force mate) with almost any combination of material. The only exceptions I can think of are K+B vs. K, and K+N vs. K. In that case I don't think it's possible to mate at all. To the point made in OP's post, does chess.com consider K+N vs. K+N to be insufficient material? If so, then it's wrong -- clearly that is sufficient material to mate (although not to force a mate).

It quite rightly doesn't award the win with King and Two knights vs K either, even though the mate could obviously happen (but not be forced)

I've had knight vs knight in a bullet game quite a few times and it doesn't automatically declare the game drawn until one player's time has expired.
So I think in the given situations you would undoubtedly get the mate as it would definitely allow the final move to be made.
The problem that you may have is that if the opponent realised what was happening is they could just wait for the time to expire rather than making the move that would allow mate - that would enable them to get them draw.
That isn't a problem here- if white captures the queen, black mates with the knight immediately; white doesn't have time to let his clock run down. However, it can be a problem with trying to play Stamma's mate.
I've been trying to invent a position from which White can force mate, except Black would get a draw due to the minimum material rule. I haven't been able to invent one. It makes a good puzzle!
To the point made in OP's post, does chess.com consider K+N vs. K+N to be insufficient material? If so, then it's wrong -- clearly that is sufficient material to mate (although not to force a mate).
Well, this is believed by many:
http://www.netplaces.com/chess-basics/ending-the-game/insufficient-mating-material.htm
"A king and knight cannot checkmate another king and knight..."

O yea I didn't even notice that could happen, my friend had that experience in a similar position and it said game drawn- insufficient material or whatever it says when neither player can mate

The insufficient material rule does not cover King + Bishop v. King + [opposite-colored] Bishop, because mate is still possible, as demonstrated.
The insufficient material rule does cover King + Bishop v. King + [same-colored] Bishop.
I don't know how the site adjudicates King + Knight v. King + Knight, but the insufficient material rule does not cover that material configuration either.