1999 Sunfire problem

Hi, I am having the following problem with my 1999 Pontiac Sunfire (2.2L, automatic):

When I try to go at constant speed or accelerate very slowly the car backfires…. When I accelerate more firmly (about 50% of the accelerator range or higher) or leave the accelerator alone so the car can go with its inertia, the problem goes away…. when i leave the car on parking or neutral or in a gear without moving it works perfectly....

I have some problems with the muffler that I will be taking care in the following weeks, but if the problem is that will it go away when I accelerate harder or stop accelerating??? (I mean, can I rule out a muffler problem based on the conditions of the problem??)…

Somebody told me it can be the transmission or the fuel pump, but the problem also happens when I drive on 1st, will that rule out transmission issues or it might still be present even when it’s driving only in first gear???

About once a month the cart doesn’t want to start and only does it when I push the gas, does that indicate something related to the fuel pump????

I changed the spark plugs, the wire set, the fuel filter, air filter and cleaned the injectors…

I know it will be almost impossible to locate the problem with an email, but I want to know if there are tricks or tests I can try to rule out some areas of the engine instead of changing parts that doesn't need to be change (the muffler needs to be changed but for example the fuel pump might be ok)...

Any help will we most welcome….

Thanks…