American football, forward pass after catching the ball on a kickoff?

Not allowed. It is an illegal forward pass and the kick return team is called for a penalty. However, you are allowed to throw a lateral, in which the person receving the football from you is either parallel to you or behind you.

A forward pass is illegal after any change of possession including a kickoff. The penalty is 5 yards from the spot of the foul. Just like a legal forward pass, if it is incomplete then the ball is dead. If for some unlikely reason that I cannot fathom, the penalty was declined, the end of the run is the spot the pass was released.

If this happened on the last down of a quarter, the quarter is not extended because an illegal forward pass normally carries a "loss of down" as part of the penalty. Although the loss of down provision doesn't apply to after a change of possession, it still affects this situation.

If the passed ball is not caught, it is ruled an incomplete pass. The penalty is 5 yards from the spot of the foul (the point at which the ball was thrown). and a 1st and ten for the offense.

It is illegal in all levels of play.

On a change of possession, the only pass that is ALLOWED (not 'aloud), is a backward pass. There is no such thing as a "lateral" pass, only forward, or backward, passes are recognized in the rule books. "Lateral" is a term the morons in the booth use. Sarge

no forward pass is allowed in that situation. if a forward pass from the QB to a RB or WR is dropped then it is a dead ball, a backwards pass or in-line pass would be considered a fumble.

it only can go backwards and if missed it is a live ball

if they toss it forward it is a 15 yard penalty from the spot of the toss