If a receiver catches the ball and then comes down with the ball. He then injures himself during the play. While going down he tries to lateral the ball backward. If there is no teammate nearby and a defender catches the lateral is there a rule that states that if the player is giving himself up due to the injury the play is dead at the spot of the injury with possession remaining with the offensive team?

No, there is no rule at any level that makes this kind of statement.

I think in pick up basketball, if you become hurt during a play in which you lose possession, it is common courtesy for the ball to remain in your team’s possession.

This happened to Doug Williams in Super Bowl XXII. He went back to pass and when he planted, his knee gave out. He let go of the ball, and Denver recovered. The officials gave the ball back to Washington though.

Nice mistaken description. On that play, it was ruled that Williams was throwing the ball when he went down - incomplete pass.

Hmmmm...not the way I remember it being described (or how it occurred), especially since the ball went backwards, and he was not attepmting a throw when he went down as I recall. I remember language to the effect of Williams was giving himself up because of the injury, so the officials called the play dead, and the Redskins retained the ball. I could be wrong, and if so, I apologize for my "nice mistaken description" :-)