"You don't have the guts to do that." = You don't have the courage to do that. guts = courage, bravery, mettle "Spit your guts out." = Could either mean "confess" or "vomit" - dependent on context.

guts = courage

Your "guts" are your internal organs. Mainly the stomach and intestine, but it could mean all internal organs. "To have guts" means to have courage or boldness. Cowards are called "gutless". "Gutsy" is sometimes used as a colloquial synonym for "bold". A poetic-sounding equivalent is "to have intestinal fortitude". I'm not sure how "spit out the guts" fits (do you have some more dialogue?), but Padraig's explanation is clearer. Maybe "spit out the guts" could mean "tell us the most important parts". The internal mechanics or circuitry of a machine is casually referred to as "the guts". When someone is extremely disappointed (even upset), they are "gutted" (as if they were a fish, and someone pulled out their innards - so a feeling of hollowness is left). "I'm gutted that my team didn't get to the finals this year."

"You have no guts to do that" means you have no courage or determination to do it, especially if it is hard or difficult to do. I'm not sure about the meaning of the second phrase whether it means "vomit" or something else, but I know that the phrase "sweat one's guts out" means to work very hard.