Suddenly I am very curious if you find the existence of photons equally offensive.
Because, like, a lot of the appeal of gravitons as an idea is that all the other fundamental forces have a particle associated with them; photons for electromagnetism, W and Z bosons for the weak interaction, gluons for the strong interaction.
Also, to take a stab at answering your questions: no, you can't run out of exchange particles. There isn't a finite supply, new ones are created and destroyed all the time.
Yes, there's a lightspeed delay on gravity. It would be weird if there wasn't; if the effects of gravity were immediate you could theoretically use that for FTL signalling and break causality.
(guessing here) I think gravitons can interact with each other, but you shouldn't think of it as tiny balls crashing against each other and bouncing off in whatever direction. Fundamental particles are screwy and do not behave like macroscopic objects. Also, their only interaction would be gravitational anyway, and gravitational interaction is super weak at that scale, so they wouldn't interact a lot.