Why do the warmest days of summer usually occur after the longest day?

Answer 1

For the same reason the coldest days of winter come after the shortest day: it takes several weeks for the earth to warm up or cool down because the soil, the water and the atmosphere are all very massive and tend to maintain whatever temperature they happen to be at. You see the same effect when you put a pot of water over an open flame - the water heats up slowly, not instantly, and if you put a hot cup of water in the freezer it takes time to cool down.