I see multiple issues here, but the pushrod popping out is simply a lifter bleeding down. As CSK mentioned, you can run the pushrod .020" off the bottom of the lifter, and it will eliminate that problem.

That could open up another can of worms though. If the motor goes into valve float, the lifter will do it's job and pump up. That will hold the valve open and if P2V clearance is tight, could bend valves. Looking at the sweep pattern on the valve tip, I'd say the geometry is pretty far off and instability is likely. Also, it appears the adjuster is out much further than it should be. The preferred setting is one turn out from seated in the rocker body, and no more than two turns. That's turns, not threads.

Something else that is misunderstood is that valve float does not limit rpm. The valve being off the seat when it is supposed to be closed is what limits rpm. If valve float did it, lift rule class racers that run lofting cam lobes couldn't turn any rpm due to the controlled valve float. A hydraulic lifter does not really get into valvetrain instability any earlier (or at least not much earlier) than a solid lifter. What it does do is hold the valve open when the lifter pumps up from float, which then holds the valve off the seat. A solid will go into float and jackhammer the valvetrain into oblivion, but won't stop revving until the instability is so severe that the lifter bounce causes the valve to bounce off the seat and rpms stop climbing. Either way, the instability is present and the valvetrain is not happy. The preload has nothing to do with rpm potential, and the only reason for a zero preload hydraulic lifter, with a heavy retaining clip, is to allow the engine to continue revving like a solid when valvetrain instability is present.

Do the best you can at getting the valvetrain stable, and you will improve rpm potential regardless which lifter style is used. I didn't say they would both be equal, but improved.