Disclaimer: I'm not a car expert, nor a mechanic; I just have this car and fixed a similar problem. I recommend you read my story, see what fits, and apply appropriately.
Keep in mind, by the way, that most of this will quickly exceed the value of the car unless you can do the work yourself.
A few years ago, my Lumina had a niggling problem; little pauses when coming off stop signs. Nothing serious, and very very occasionally: it would die when I put it in gear or hit the gas from a standstill. No one could ever figure out what it was, so we let it be; not a big concern.
About a year later, the engine suddenly started stumbling a lot, so the injectors had to be replaced, which I did (I used rebuilt). The car returned to the above symptoms, which really weren't a problem.
A year and a half later, the "dies when put in gear" got *suddenly* worse, and I had to start "priming it" -- which is a bad idea in these engines; you can seriously screw up your cat -- and then it became "always have to double-pedal" when slowing down/stopping. This included cruising on the highway. It got progressively worse, until the engine started "searching for idle" hard starting, and dying if the RPMs were allowed to settle. All of this was MUCH worse if the weather OR the engine was warm (making it *very* hard to start it again, leading to the occasional discovery that hitting the gas slightly helped). It also started dropping oil--which I later found was a mistake made by a "fast oil change shop"--and I stopped trying to run it when I heard a new, sharp ticking sound and it just wouldn't stay running.
I spent some time testing the car's computer (made my own tester with $12 or Radio Shack parts and an old laptop) and found nothing wrong--O2 sensor was ok (oscillating within range), temperature ok, no codes other than the one caused by me holding down the gas as well as the brakes all the time (the paper clip check can reveal your trouble codes IF YOUR CAR IS THE RIGHT YEAR, i.e. old enough--better check!). I tried disconning the airflow sensor, checked the IAC, checked for vacuum problems (does it stumble when you hit the brakes, does ?carb fluid I think? sprayed in the right areas cause a change in how it runs?) and found several articles that said--in this engine--sometimes just cleaning carbon out of the throttle body fixes it. I couldn't get to the TB so I stopped poking at it for the winter.
A few weeks ago I jumped the car with a friend. After doing this and starting it, my friend noticed a 6-inch spark
jumping from my shortest plug wire. This is a COMMON PROBLEM with these
cars! We replaced the plug wire and the car returned to
searching-for-idle-doesn't want to stay running, but the battery was fried. It had leaked/grown crystals/shorted to the inside chassis (check this, apparently the battery can be too close near the front), so we replaced it. Now I had a car I could drive out of town.
After a few quick updates: replacing all the plug wires (we only did the clearly bad one, to relocate the vehicle), fixing a radiator fan that (I'm not sure) had died, changing oil, other regular maintenance, I still had the trouble-at-stop-signs problem. This turned out to be a (an idle?) fuel-pressure regulator. This was *expensive*--as was the brake work I had to do because I double-peddled for far too long--but the car (which is very low miles and in excellent shape otherwise) now runs like it did when I got it. It died twice after replacing this--both times at stop signs--but this is because the computer (which had been disconnected from the battery) was re-learning idle settings AND I needed to run injector cleaner. It no longer dies.
Based on my experience--and all of my research, though I'm not a mechanic--here's what I think:
Don't pump your gas! If you dump gasoline in your catalytic converter (or, e.g., "top off" your fuel tank to get that last bit in), you can destroy it. You may be backfiring because this is what's happening anyway...unburned gasoline...so I'd check plug wires. Loss of horsepower could be a plugged / damaged cat...bad wires...throttle body...injectors...fuel filter...fuel pressure...and on, and on. Unfortunately, you might be looking at a lot of cash (or a $14 filter), a lot of elbow grease (or a quick plug wire change). My problem was mostly fuel pressure...but I had power with rpm's. Yours (with loss of power) might indicate backpressure, and some of your symptoms might be explained by flooding the cylinders. Finally, I'd check one other thing--with the car having been off a while, in a quiet area, open your driver's-side door, put the key in and turn it to "on" (not "start") and listen for the whirr-click of the fuel pump priming the injectors. If you don't hear this...you have a place to start. I only heard (or rather, didn't hear) mine fail a couple of times--but it was enough.
If you get it running, I was given one last bit of advice: since the injector cleaner solved the remaining stutter, I was also told to stop using 89-octane (ethanol-added) fuel, and instead use straight gasoline.
I can't comment on bypassing the heater core.
Source: own experience, same engine