I do feel it's best to let somone fix it for you but maybe not. Anyhow,here's where the problem seems to be. YOU STATED "Also when is the gas exhaust fan supposed to come on? I think in one try it did not turn on. Later I fed power directly to the motor but it did not help." The "gas exhust fan" is known as induced draft. If it didn't run when "fed power directly" the motor should be replaced. My reluctance was for good cause which is recconizable to one who works in the trade. Far too much can be lost in translation while dealing with flamable gas or electricty. Dealing with both at once is invitation for disaster. Here is an example. When you said "Later I fed power directly to the motor but it did not help." That could mean the draft motor still didn't run,,,, OR,,,,,it could mean the blower motor still didn't come on. No harm,no foul but such way of comunicating is not good while working on gas and electrical. Engineers pride themselves on being exact so you should understand where I'm coming from.

What you describe certainly indicates the start capacitor is bad. Is it a round black cylinder shape? Make sure the power is turned off. Look at the end where the terminals are hooked up and you'll see the black goo came out of a round hole. If it weren't for that hole, the capacitor would explode. If you are going to replace that capacitor you may want to replace the start relay also. Although I have seen times when just the capacitor went, most times it's the fault of the relay. The start capacitor can only stay in the circuit for about 7 seconds and then it will blow. It is made differently from the run capacitor which can stay in the circuit indefinitely. So if the relay points have a tendency to stick or are fuzed together, it'll take out the capacitor. I can't quote prices because they range wildly depending on area of the country and from supplier to supplier. In my area, the relay would retail for about $15.00 and the capacitor for about $8.00. Make sure you replace it with the same MFD and voltage rating. It should be stamped right on the capacitor. Take a look at the fan and compressor run capacitor. If it is swollen or leaking, you'll want to replace it too. If you have a capacitor checker or know how to check capacitors, give it a check. Otherwise, just for a margin of safety, since you're already in there, replace that too.

Not an expert but sounds as if while the burner is on the flue gases are going up the chimney. When the burner cuts out the chimney effect quits and the blower is producing a negative pressure on the chimney or a leaking heat exchanger pulling the exhaust gases into the system. Again no expert but a leak is indicated. I think the CO detectors will not detect until a minimum threshold. The limited amount of time to possible CO exposure after the burner cuts out means the detectors may have insufficient CO present to set off alarm.

As others have said ... the behavior during cooling depends on the individual system and how options that may be available are set. The first two houses and one apartment in which I've lived had systems that shut the blower off immediately. My current house (built in 2004, Carrier system) runs the blower for an additional couple mins. All the systems at the business where I've worked (three systems 30+ years old, one 2 years) shut the blower off immediately. The coil will not freeze-up if the blower stops. Heat extraction/transfer stops when the compressor shuts off so the coil will not get colder than it is at the moment the blower stops running.