Welcome to forum. You really should post the engine make & model. Sounds like it might be a B&S Intek twin. IF so, I can send you a Service Manual for your engine IF you like. Address below, put in proper format and remind me engine model and what you want.
Sounds to me like besides the valves needing adjusted it is not getting gas as if carb solenoid is not working. You say you jumped the solenoid but you did not say that it was actually working. I would remove the solenoid, depress the plunger, wedge a tooth pick beside it to hold it depressed, break off tooth pick then re-install solenoid and see if that helps. That way you have not ruined the solenoid. You say it runs 10 seconds, it would not run that long with one shot of starter fluid. Don't do it anyway as it will ruin your engine.
Walt Conner
wconner5 at frontier dot com