I'd get that evap solenoid fixed first. Normally, such solenoids open when the engine is warmed up and running so intake manifold vacuum can draw gas fumes out of the charcoal canister. If your valve is stuck ON, or leaking when it should be OFF, then that is similar to a vacuum leak which makes started difficult for pretty much any car.
When you add gas, you are opening the fuel tank so any vacuum that accumulates in it, as fuel is drawn out by the engine, gets lost. Normally fuel tank vacuum builds up a bit, then the gas cap's spring-loaded valve opens for a moment to let fresh air in, reducing the vacuum. The computer monitors for this pressure cycling; when it doesn't see it you'll get OBD-II codes. When you open the gas cap and eliminate ALL vacuum in the tank, the leaking evap solenoid can let air from the fuel tank & charcoal canister (evaporative emissions system) parts into the engine, resulting in low vacuum at start-up. Basically, rather than drawing air through the air filter, throttle plates, etc. (i.e. the normal way) the engine is "breathing" through the evap canister. Once you've driven a bit, vacuum collects in the gas tank again so there is less air in there... the engine has to "suck air" through the normal path: air filter, throttle plates, etc. past the sensors that the computer uses to measure airflow.
I'd expect just opening the gas cap - without adding fuel - would cause the same symptoms. Try it as a test. If the car starts properly after opening the cap, then the bug is something else... like gas getting into the evaporative emissions control system flooding the engine. Holding the throttle wide-open helps because you're adding a lot more air this way. Over-filling the tank can let gas into places it's not supposed to be...
mike c.