Is the quarterback allowed to make an underhand throw in the NFL? If so, if the pass is incomplete is the play dead?
From the NFL rulebook (Rule 8, Section 1, Article 1), the definition of a forward pass is when
There is no distinction made between overhand, underhand, or sidearm passes and they all follow the same rules..
so yes, an incomplete pass would warrant the play being dead.
There are plenty of instances of underhand passes being used in the NFL and other levels of play, usually as a shovel pass (a kind of screen pass), but also sometimes when the quarterback is being dragged down and wants to get rid of the ball as quickly as possible.
The quarterback may throw the ball in any way he likes as long as he uses his arm(s). Whatever way he decides to throw the ball - underhand, granny-style, basketball shot-style, in between his legs, whatever - they are all treated exactly the same.
If he throws it underhand forward, that is perfectly legal. If the receiver drops this the ball it is dead, incomplete pass.
Now let's go ahead and add two other examples that we are seeing a lot now.
1) Can a quarterback throw overhand backwards? The answer is still the same. Any ball thrown overhand, underhand or whatever backwards is all the same. The difference on a backwards pass is that the throw isn't actually a throw, it is treated the same as a hand-off. Meaning if dropped it is a live ball. Also for stats purposes if a QB throws the ball backwards 20 yards across the field to a WR, it is not a passing attempt, it is a rush attempt by the WR. Several times this year in the NFL I have seen scorers count these overhand backwards passes as passing yards/receiving yards and watched them get changed to rush attempts.
2) Some coaches have (IMO) used the passing rules to manipulate fumble risks and passing/receiving stats for their team. In your basic spread formations you have several variations of motion or jet sweeps. It has become the trend for the QB to use a pop pass in these cases instead of a hand-off. The pop pass is simply the QB releasing the ball in the air with two hands right in front of him. If you actually slow down most pop passes the ball indeed travels backwards but in real time looks to be even or the same. The reason the ball travels backwards is because as QB receives the ball in a gun formation he is going backwards (even if it is slightly). So when jet receiver drops this pass/hand-off it is ruled an incompletion. When he catches it it should be statted as a completion for however many yards the WR gets running. This allows a QB/WR to rack up stats against "slower" competition while proving no skill set. (I watched tape on a kid this year with 1200 yards receiving at the high school level - which if you don't know is excellent. 500+ of the yards where on jet sweeps. Still 700 yards is great and he is a great prospect but very very deceiving.)