Sounds like it could be a lot of things, so I'll go down the list:
First, make sure the small butterfly valve at the lower right side of the tank is fully open.
Your fuel filter should be full, not just fuel in the bottom. I'm assuming you have a glass bowl. If you see any bubble or air in the bowl, your filter is not fullof fuel. This could be air entering the pickup line anywhere from inside the tank to, and including the, the fuel filter, or a bad lift pump. Pull the pickup tube on the tank and check that it's there and tight. Disconnect the line before the lift pump and pump the squeeze ball, fuel should come out. Disconnect a line after the lift pump and see if fuel comes out when you crank it. If it does, pinch the end slightly to see if you have adequate pressure as well as flow. Disconnect a line after the filter to see if the filter is clogged, or bubbly to see if it's sucking air at the filter.
Your throttle rod should not move on its own. It is connected to the lever in the cab, and held in position by spring pressure at the hand lever's pivot point. At the engine end, it is linked to the throttle arm which is stopped by adjustable screws with locknuts at high and idle.
The fuel shutoff lever on the engine should be against or right close to the stop when the engine is keyed off, and when keyed on, should pull over and be against or right close to the opposite stop to run. If it's somewhere in the middle, it may be in a partially shut off position, and won't run well if above idle or needing more hp.
The solenoid should click over to run when you key the engine to run, and let go to stop when you key off. You will hear definite clicks when you do this. Some engines have an arrow with the word "Stop" and some the arrow faces opposite and have the word "run", but in either case, the solenoid should pull to run and let go to stop, and again, the lever should be fully in these positions. Any where in between and the engine either won't run well or shut off reliably.
You want to make sure your right steering lever is not pulled(pushed?) over and locked into aux detent. This loads the engine and will kill it unless you have an open-center attachment hooked up to the aux couplers.
Check for unrestricted air flow.
If all this checks out, you may have an injection system problem.
Good luck.