Fuel primer bulb does not stay hard when engine is running, rather it drops to about half full (normal). Your post is a little vague.....
Are you saying that the engine hits full throttle okay and runs okay at full throttle for 3 or 4 minutes, then drops rpms?
Warning horn should beep when the key is turned on.... does it? To check warning horn circuit, have the key in the ON position, then ground out the TAN wire (overheat sensor) that you'll see protruding from the cylinder head. The horn should sound when that wire is grounded... if it doesn't, find out why as that's the only warning system you have if the engine should start to overheat.
Start trouble shooting with the basics.... check compression.... what are the readings? With the spark plugs removed, check for proper spark which should jump a 7/16" gap with a strong blue lightning like flame, a real SNAP! The gap is important!
What spark plugs are you using and at what gap?