May a Cessna 182 land on a major, high-traffic airport, say EDDF (Frankfurt)?
There is the General Aviation Terminal (GAT) in Frankfurt that is handling "executive and private aviation flights" as they say. General aviation airplanes and helicopters are handled at this GAT that is open 24 hours per day. I think the best is to contact them.

Some additional rules related to GA in Frankfurt can be found here. Bus service is mandatory for passengers between aircraft and GAT. The list of services includes a sentence Lashing of small aircraft. So probably some not very big are expected to arrive.

Landing fees in this airport are used solely to refinance airport infrastructures and the resulting operating expenses. They are regulated, and, while not exactly very cheap, they are not astronomic either.