65 Barracuda - Idle and no start issues

Did you just run the battery down causing the no crank condition? As for the dying, It could be a number of things, but, sounds like fuel to me. Maybe that carb has a problem, not adjusted right, flooding, vapor lock, etc. Just because it's "new" doesn't mean it's working correctly. You'll need to diagnose it at the time it acts up, not wait a few of days. You may have to drive it around the block a few times to get it fully warmed up and off the choke. Try to recreate the same conditions in your driveway so you have the tools there to work on it.