Accidentally put one quart of transmission fluid in my engine how much damage am I causing driving the car
ATF is glycol based. Yes, there are many oils used in ATF under careful and exact chemistry. Oil is oil based. (fancy that). Either can be synthetic. Oil lubricates under very extreme conditions. ATF is meant to transmute power within the liquid. It cannot hold up to extreme temperatures. Mainly, when two dissimilar liquids are used in an engine or piece of equipment, they cannot and will not mix. Little air bubbles are produced. The air bubbles provide no lubrication. Therefore, it's like running your engine with no oil at all. The same thing happens if you run two different types of oil, and sometimes (rarely) even if you run two different brands. Change your oil immediately. Don't run the engine until you do.
I've been using about one quart ATF (10-20% total oil capacity) in my cars routinely for 30 years and it has only helped the engines, both in the crankcase and the fuel tank (add 0.5 ounces per gallon of gas). I've used it in late model, low mileage, high compression Mercedes Benz, Fords, a Dodge RAM 1500; my 5 hp generator, a couple of lawn motors. It greatly improves rough idle, engine misfire, highway ping and sagging compression; frees and quiets valve train ticks, removes sludge, stops leaking seals, decarbonizes and frees fuel injectors, valve stems and faces, EGR valves and more. It restores pick-up within minutes, after a couple of wide-open-throttle blasts on the highway. On old, high mileage engines, use 2-4 oz. per gallon for half a tank. Spray mist lightly through the throttle body at high idle for immediate improvement (will not hurt MAF sensors).
There seems to be a lot of misinformation regarding ATF. It's a robust blend of petroleum oils (not glycols, like coolant) and, as such, acts as both a hydraulic fluid and an excellent lubricant. It's approximately SAE 20 grade and will lower crankcase viscosity at most 3 points (e.g., 10w-40 to 10w-37). There are no detergents in it like phosphorus or magnesium. It contains no polmerizing seal conditioners, so it won't swell any gaskets or seals. It's highly refined, with a low sulphur content, so when it burns it leaves no ash or black soot. It contains no heavy metals or zinc, so it won't damage catalytic converters when used in the gas. Improves oil flow, enhancing warm-up and heat distribution throughout the engine. It's used widely in industry and is found commonly in power steering pumps and a/c and refrigerator compressors. See here, for example, as a reference.
The principle advantage is that it softens (and eventually dissolves) carbon, sludge, varnish and lacquer. These buildups prevent motor oil from penetrating and lubricating vital engine parts, greatly accelerated wear. I see this problem constantly, even in my well maintained engines using full synthetic oil. Gummy build-up in the fuel system wreaks havoc on injectors and thus the engine's fuel-air ratio (top tier gasoline just isn't enough, especially in winter months). Deposits also cause parts to stick together - such as piston rings, hydraulic lifters, valve faces, PCV valves and rockers - and block oil passages throughout the crankcase, variable valve timing mechanisms, and the oil pump. The leading cause of gasket and seal failure is acidic sludge; ATF removes sludge and thus prolongs gasket life. Over time, ATF will pass into the "dry" side of the exhaust, cleaning the EGR system and the face of catalytic converters. I have seen these benefits first hand, on every engine I've used it.
There are some minor disadvantages to using ATF. It will slowly break down motor oil itself, so be sure to change crankcase oil every 3,000 miles. Carbon removed from piston rings and cylinder walls may actually increase the oil consumption on an old, high-mileage car (I've compensated by using Engine Restore additive in the crankcase). Dislodged carbon particles will at first accumulate in oil filters, causing a slight oil light flicker on the dashboard until the oil filter is replaced. Carbon bits will also occasionally get lodged into seals (like the rear main seal), causing leaks temporarily; continued use will clear the seals and stop the leaks within a few months. Improved compression will also increase intake manifold vacuum, enhancing existing vacuum leaks, so much as to trigger a CEL light; the remedy is of course to find the vacuum leak, typically around the PCV valve & hosing or the gas cap.
Do not listen to anyone who says its ok or dont run it long and change the oil asap. Trans fluid will destroy your rear and front seals. At first it swells them but in a period of about and hour or two after running it the rubber in the seals will become mushy and within a day or two youll be missing oil as it begins to leak from the front and rear seal. As an example find a piece of rubber and put ATF on it and check it the next day. The rubber will fall apart in your hands. Rubber and ATF never go together.