1989 Acura Integra Engine has trouble starting and idling on

Engine problem
1989 Acura Integra Front Wheel Drive Manual 313600 miles

A few months ago I took my car to a shop to get a coolant flush. Ever since the morning after I drove my car home that day I've had problems. A few days after I first got the car back the car would crank right up on cold starts but it wouldn't idle unless I applied a significant amount of throttle. When I would start the car without applying throttle, it would immediately shut off and I'd have to crank it up again. To get the car to idle, I'd have crank it up and push the accelerator all the way to the floor. During this time, the engine sputters and backfires for a few seconds with the rpms dancing between about 200-1100. If I lift my foot off the gas during this time, the engine immediately stalls. Once the revs finally manage to get up to about 1250, it seems like whatever was holding the engine back "releases" it and the engine runs smoothly. I have to quickly ease off the gas to prevent the car from redlining and sit there applying a moderate amount of gas to keep the revs up at about 2000rpms or more to let the engine warm up, then gently and slowly ease off the gas and "guide" the engine back down to idle speeds. Whenever I abruptly let off the gas when the engine is not warm enough, the RPMs drop to zero with the engine backfiring and shaking on the way back down after getting below about 2000rpms and the car shuts off. When the engine is warm, the RPMs drop to zero and the car dies, but there is no rumbling or backfiring involved.

On warm starts, the car cranks up on the first try but barely stays running at idle when I don't give it any gas. Occasionally it will stall in that situation. It also shuts off if I abruptly let off the gas, but it isn't guaranteed to happen as when the engine is cold. It runs fine once I actually get going and when on the highways, but the engine has problems idling whenever I put it in neutral (I drive a manual). Also, when I do manage to get the car to idle on its own after I've "guided" the engine back down to idle speed, the rpms will fluctuate for awhile, i.E. The tach needle bounces up and down anywhere from about 200rpms to 900rpms. Over these last few months, all of the above conditions have worsened, which I suspect is due to the weather getting colder. Where it used to crank up on the first try, now it often takes me 7-8 cranks before the car will stay running: About 2-3 tries where the engine won't start at all, 3-4 tries when the engine will start for a split second then immediately die, regardless of if I have the gas floored or not, then on about the 7th or 8th attempt it will stay running when I floor the gas, albeit it has all the problems mentioned above. The check engine light has also been on ever since the morning after I got it back from the shop. I've also noticed bluish smoke and a mixture consisting of oil and what appears to be water leaking from the tailpipe. There is also a leak under the engine, but that's been there for years. There is also a whining noise coming from the left rear which I've been told is the fuel pump. However, I've heard this whining noise for years, and I didn't have these problems described above until a few months ago.