Do a voltage drop test. It would be important to do this before doing a compression test
Specially considering it "pops right off when pushed and put in third gear"
This tells me the starter is sucking too much voltage away from the fuel solenoid...
Can you elaborate on this starter sucking voltage from the solenoid scenario? I think I may be currently experiencing a similar problem!
Same thing as the voltage drop thing other members have mentioned.
All the power available in the electric circutry (note I didn't say battery) is being used to spin the starter.
Yes, you see 13.5 volts at the battery, pick a spot (cigarette lighter) and you will see 13.5 volts but put a load on such as your starter and it will draw the voltage down far below what would be necessary to run the fuel solenoid.
This is usually caused by corroded electrical contact points mostly at the battery introducing a high resistance in the electrical circuit.
I was just faced with a similar issue with my samurai. I would turn the key on, I would get the glow plug light on, everything was functoning as it should. I turn the key to start I would get a *snap* and all would go dead.... 5 minutes later, head lights would burn bright, glow plugs would work I would turn the key to crank *snap* all would go dead again...
I grabbed some steel wool, buffed up the posts on the battery, buffed up the terminals that go on that battery secured everything nice and tight. Problem solved!
I had the exact same scenario you have explained but with a boat motor. It would crank but without a start. At that time I dismissed the notion it was corroded battery terminals because it was cranking. After stubbornly checking everything else and wearing the battery down some with the cranking, I cleaned up the posts and the darn thing fired right off...
Grab yourself some steel wool. Start at the battery and work your way back on both the ground and the hot side. Make sure there is no tarnish at any of the connections and they are properly secured.