Is American Airlines super strict on carry-on dimensions for luggage?

It will fit. They only put those dimensions on the carry on rule to ensure that all the bags will fit. However, it you try to stuff it in and it doesn't fit, then be prepared to gate check the bag. American also charges for the first bag, so they may treat it as a checked bag if they make you gate check it. American Airlines at the airport I work at does this all the time!

American Airlines Carry On Dimensions

When you go to put your bag in the overhead compartment it will not fit. The door will not close. My 22 inch including the wheels just barely fits. Then you will be required to check the bag.

I doubt they will say anything. I always see people with HUGE carry-ons that I know dont fit into those little things before the gates. Then they MAKE them fit into the overhead compartments. I would just TRY to not over pack it. So if you have to put it in that little thing before the gate you have some room to work with to squash it into it :)

Not really, it depends on the plane, how full the flight is and the mood of the flight crew.

Go ahead and bring it -- worst case they gate check it.

It varies based on one person to the next.

if you board early and are super polite you should be fine. If you are surly expect a gate check

yes they are, and that would be so terrible to have to check a bag.

no