Suppose the max crosswind rated for my C-182 is 15kts, but the wind is directly across and gusting to 20kts. Is it legal for me to go? I've heard plenty hangar flying stories of many an operation that exceeds the design specifications of certain planes that end successfully.
The 15kt crosswind for the 182 is not a design limit at all. If it were a design limit, it would be specifically stated in the POH. To be honest, I think there's no such thing as a design limit for crosswind. It all comes down to how much rudder authority there's left at a specific crosswind strength. And even that depends on many factors like approach speed, flap settings, weight, CG, etc.
This is what the POH for the C182T says:
So it is perfectly legal to land and takeoff with a stronger crosswind component. It's up to the pilot to determine if it's safe to do so or not. It all depends what the pilot's capabilities are.
Not if it is actually a design limit. In the case of the 182 it probably isn't actually a limit though.
With regard to crosswind information provided by the manufacturer, there are two different types. For example, here are excerpts from the Falcon 50 AFM (which I happen to have handy):
Demonstrated Crosswind Component
A demonstrated crosswind component is just that. Demonstrated. It is the highest amount of crosswind that was encountered during the certification process and is put into the AFM as a guide. It is not a limiting factor though. This is what the 182 has in its POH.
In this case, we have an actual crosswind limitation which can not be legally exceeded. (Note that in this case this is for a one engine inoperative ferry.)
In the case of the Falcon, it is certified under 14 CFR Part 25. The requirements are similar (but not the same) for Part 23 airplanes. In particular, we need to look at:
This basically says that they need to test it up to 25 knots. They can choose to test it higher if they want to, but it isn't required. If they find that there is a speed less than 25 knots where the airplane isn't "safe", then they need to impose an actual operating limitation. Otherwise, they just publish the value that it was tested to for reference and it is not a limiting value.