Hasnt it got something to do with tornadoes? Cats and dogs were swept into the tornado and when it had passed they rained down on the unsuspecting people. just a guess (it worked with fish and frogs)
Ah, the world of Forteana! Its possible as a derivation. Herodotus recorded rains of fish, and certainly Charles Fort recorded rains of Cats, Dogs and Pitchforks. The mechanism for these is unclear, though.
As a source for the phrase , well, maybe. I wondered if it might have to do with the noise the rain made on a roof?
In the 1500s, houses had thatched roofs. Thick straw, piled high, with no wood underneath. It was the only place for animals to get warm, so all the pets... dogs, cats and other small animals, mice, rats, bugs lived in the roof. When it rained it became slippery and sometimes the animals would slip and fall off the roof. Hence the saying, "It's raining cats and dogs,"
Edinburgh is built on a hill. At the bottom of the hill there used to be a loch, and all the streets slope down towards it. When it rained the water used to flow down the streets, and if the rain was heavy enough it would wash all the rubbish from the streets down into the loch. Cats and dogs frequently died in the streets, and they too were swept downhill by the rain. People used to see them being swept past their front doors by ther rain, and they would say "It's raining cats and dogs" meaning it's raining heavily enough for dead animals to be swept away by the rain. I was told this by a historian who runs guided tours of Edinburgh.