Does the game end the instant the flag falls?

Not by itself. Art. 6.8 says the flag has fallen when the arbiter observes it or when a player makes a valid claim, so a game can carry on for several moves past zero if neither player is looking. What does end things instantly is checkmate: Art. 5.1.1 ends the game the moment mate is played, so mate delivered before the flag was claimed wins.

What if both flags fall?

In a game with an increment this hardly ever happens, because a flag that falls is usually noticed within a move. If it does happen and nobody can establish which fell first, call the arbiter rather than arguing: the answer depends on the format and on what the arbiter observed, and the players are the two people least able to judge it fairly.

What happens when your time runs out?

Black is a rook up with checkmate available in one move. Black’s flag falls. The result is half a point each, and it is not a mistake.

Black to play mates at once with 1...Rd1. If the flag falls first the game is drawn: White's lone king cannot mate by any legal series of moves.

The exception, word by word

Art. 6.9 says the player whose time runs out loses the game, then adds this: the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.

Two words carry the weight. Possible, and any. Not forced, not likely, not sensible. If some legal sequence of moves ends in mate, even one that no human would choose, the flagged player loses. So, going by what your opponent has left on the board:

A lone king cannot mate anybody. Draw.
King and bishop, or king and knight, cannot mate either. Draw.
King and two knights can mate, because a defender is allowed to walk into it, even though it cannot be forced. You lose.
King and one pawn: you lose, because the pawn can promote to a queen.

This is a stricter test than the insufficient material rule that ends a game on its own, and it is the reason a website sometimes gives you a draw where you expected a win.

Somebody has to notice

A flag counts as fallen when the arbiter observes it or when a player makes a valid claim (Art. 6.8). In a slow game with arbiters walking the hall, it gets picked up. In blitz, assume nobody is watching your board.

So if your opponent’s time is out, stop the clock, say so, and put your hand up. Do not play another move first and do not sit there being polite about it. If a control has a move requirement, such as forty moves in ninety minutes, the check is whether those moves were completed before the flag, which is one more reason your scoresheet needs to be up to date.

Online it is done for you

Servers flag you at the instant of zero and award a draw when the winner has no mating material. Each site codes its own approximation of Art. 6.9 and they disagree at the edges, so do not learn the rule from what your app did last Tuesday.

The habit that saves the most half points is glancing at the clock on the same beat as the board, every move, not just when it feels tight. The skill that saves whole points is finishing fast, and that is trained rather than willed: people who spot mate in two in five seconds do not lose won games on time.