It depends on which rules are used. The United States Chess Federation play by their own rules and make you pay for the rulebook. I don't know the USCF rules and have never played a USCF tournament so if it's a USCF tournament I can't help you.
Under FIDE rules for competitions at standard time controls:
1&2: Mate has to be possible by any sequence of legal moves (i.e. does not need to be forced) for a win-loss result to occur when one player runs out of time. If mate is not possible the result is a draw. Mate is possible with two knights and a king against a lone king so in this case the player with the lone king would lose on time.
3: Either player can claim a draw by 50 move rule.
4: The position prior to the illegal move is reinstated and the opponent of the player who made the illegal move is given an extra two minutes.
On Chess.com:
1&2: Material is the only consideration. The position on the board is ignored. Lone king, king and knight, king and two knights, and king and bishop are all insufficient material to win on time. The same should be true of king and multiple bishops of the same colour though I'm not sure how if that's actually the case.
3: Either player can claim a draw by fifty move rule.
4: It is not possible to make illegal moves.
I've always had a couple questions about some of the rules of chess (tournament play and online/how they might differ) so I was hoping someone here would be able to answer some:
1. If you have two knights and a king against a lone king and it is not possible to force a win within 50 moves, can you claim a draw by insufficient material (what if mate actually can be forced, would it automatically be declared a draw online)? If you run out of time with the lone king do you lose on time or draw?
2. Similarly if you have a lone knight versus a lone bishop, it is still possible to deliver mate (although your opponent would have to make serious mistakes). Is this automatically a draw? What if mate can be forced?
3. Let's say in the following diagram, White has to mate in 2 or else it will be a draw by the 50 move rule. Is it actually possible to mate in two or is it a draw by insufficient material before then?
4. If you're in serious time trouble in a tournament game and you move your king into check (let's say it's standard and not blitz) can your opponent take your king ending the game or does he/she just get extra time or something because an illegal move was made? Can you stop the clocks and call over the tournament director in this situation to add the time even if you made the mistake?
Thanks for any help clarifying!