new 86 owner

hello i just recently got myself an 84 corolla for 500. Im pretty pleased with it but it sure gonna need some work. I just got few maintenance done, like spark plugs wire set and oil change. And boy did she need a change bad never seen oil come out so thick lol. But i having some problems that i cant put my head to. For some reason my car seems to shut off sometimes, like when i got to a light it shut off when i began to brake, and sometimes when just driving normal making a turn it would just die on me. So i dont know what it is? also its a caburetor model
Another problem im getting is i can shift through gears smoothly but when im at a stop and want to get to reverse i get grinding, is it my clutch giving up already? i have tried double pumping the clutch but no works
The other thing i have a hose on the center of the pic that i cant figure out where it goes to?

thanks