Okay I read this through. I don't have much to offer. Breakers that cycle fast or get burned or smell like burning suggests a different issue than the breaker. Not a typical failure mode unless something else is wrong. Also strange that the breakers that got roasted were not in use (or low load on them).
This part seems weird:
"The dryer is on a 4 wire 220, but the white wire is tucked away as the plug is only a three prong plug."
I don't do any 220v work but it sounds like this might be the culprit. Maybe there's backfeeding on the ground circuit from the inrush on the dryer?
We ran a load of laundry last night/this morning. The dryer was running, everything was fine. I opened the door to throw something else in and hit the start button again and the living room circuit (afci) tripped. I could not get the dryer to run and the living room circuit to hold at the same time. When I had the dryer off, the 30A dryer circuit and 20A afci living room circuit held, but I was getting the voltage drop symptoms I talked about earlier (motor in two fans slowed down, lights dimmed/flickered).
When I turned off the living room breaker, the voltage drop went away.
I inspected the dryer wiring at the plug and found the ground tucked in the back of the box (4th unused wire for my 3-wire plug) may have been able to contact one of the screws on the back of the 220V outlet, there were no burn marks. I coiled it and pushed it to the back of the box and am sure it cannot touch anything now. I turned on both the 30A dryer and 20A living room breaker and both held. Started the dryer, no problems. The voltage drop symptoms disappeared.
Now this has been an intermittent problem. I have only noticed it twice (living room breaker tripped both times) though we have had voltage drop symptoms before, I just thought it was on the power company side. I don't know that the ground was contacting a live screw in the plug and it could have just been a coincidence everything started working again.
So I think it could be one of 3 things:
1. Backfeeding over the ground at the dryer plug. Which doesn't make sense why it would only happen when the dryer is on. Wouldn't the backfeeding be happening all the time.
2. The 20A afci could have a problem where it is shunting power from the hot directly to the neutral bar causing the voltage drop. These are pretty sensitive though and it being intermittent and somehow related to the dryer seems strange.
3. The dryer motor is going out and pulling too many amps on one side of the breaker. I don't think there is a capacitor on this model but I am just relating it to how an AC can dim the lights when the run capacitor starts going out and the motor draws too much current. The (Model: LE4440xWW0) dryer is probably 25 years old. I checked for parts and the Drive Motor is $125.
Do the symptoms sound like backfeeding at the dryer plug? Why would turning off the 20A breaker stop the issue?
If it happens again, next step is to replace the living room breaker, then the drive motor or dryer...