My Onan Emerald Plus Gen, 5.0 BGE-FA, will start and run normally for 5-10 min, then will start surging, and then die after about 20 seconds. This is with or without load. I changed the oil/filter and air filter last fall.

get a test light and do some tests for power and ground. some of these gens have a start switch on the gen. doing the remote start feature test will take two people to do, one to play with the switch and you outside at the gen probing. this is a good test light cheap I recommend.
you also might want to pull the remote and inspect the wires

6 12V Circuit Tester with 5 ft Lead

Yes. As the pump becomes hotter it loses its efficiency until it is not supplying enough fuel.

There is a breaker at the gen. usually near the start switch. When off no power from gen. to house.

Yes, perfectly normal. You need to hold the stop button until the generator has stopped.

check for a faulty sump oil switch . check for faulty coil ( magneto on the flywheel) as the time it runs suggests that it is breaking down internally.

Sounds like low compression. Check the compression or have a small engine shop do a leakdown test on it for you

While you didn't indicate the btus of the HVAC I assume it worked before? What you are describing sounds like the engine isn't making horsepwer. A 2800 is a small unit that a HVAC and some lights is about all it can stand. The HVAC takes about 6 x current to start it, than it does to run it. If this inrush is high and the generator engine isn't producing full HP then it will shut off and die on under frequency. I would recommend having and engine leakdown test done (not a compression test, as that doesn't tell the whole story. This will ensure all valves and rings are mechanincally in good shape, then try a basix tune up, plugs filters etc.

More likely the switch is bad, and only make run contact when in the start position.
Easy to test the switch continuity with a ohm meter when it is not running.

hi i have a BGE 3000 petrol generator it was working 1 min and stop the next it runs well but there is no power going to the panel on the front any help? thanks

The ignition coil might not be good, have u tried changing the spark plug,