The push rod is not moving up and down. When I pushed the push rod down towards the lifter and it did moved up and down but only very slightly. Push rod is not worn at the bottom.

I did adjust the rocker with the car running. The looser I made the rocker the more it backfired out the carb. Is that a sign of something?

Ill check if oil is coming out of the top of the push rod.
If the cam is not pushing the pushrod up, the valve being stuck should not be suspect yet correct? Since is not engaging the valve.
I think my next logical step should be to remove the intake manifold and take a peak at the lift and cam. Correct?