What role does the number "77" play?

At the beginning of the game, you receive a telegram which tells you to avoid the number 77. Further down the line, during the chapter, you enter a raffle in which the number you pick is 77, and ends up leading to your first "firefight".

I'm curious as to whether or not the number 77 plays a role anywhere else?

Side question: Who sent the telegram?

The number 77 is not important besides the fact that Booker picks it. The telegram is sent by one of the Lutece siblings.

Ending game SPOILERS:

That number is practically picked by Booker in all dimensions (a "constant") and that is why Robert Lutece, by using that telegram, attempts to prevent it from happening as a "scientific" bet against Rosalind Lutece.
Both Robert and Rosalind are having a "bet" on whether they can, or not, significantly change the timeline while having the same initial variables and constants. Robert believes that they can (and is losing the bet), Rosalind does not.