[Jolly Amaranto]

"Shouldn't there be a cap on the vent on the roof?"
No, what little rain that falls into the opening will not be a problem. Your septic system should handle it. Sounds like there is a restriction of some sort in the big pipe flowing to the tank. The snake makes it past it but lots of water backs up and only allows a trickle. Remove that big cleanout plug under the house and run a rotating rooter straight through to the tank. You can get these at Lowes or Home Depot. Gona have to get down and dirty in the crawl space.