No, the rain will not flow down the chimney if the cap is not installed. The purpose of a chimney cap is to keep rain and debris from entering the chimney and clogging the flue. If the cap is not installed, water can enter the chimney and damage the masonry or rust the metal components. Debris can also buildup in the flue and create a fire hazard.

It depends on the design of your chimney. If the cap is not installed, some rain may be able to get down the chimney, but it will depend on the specific design of your chimney. In general, it's a good idea to install a cap on your chimney to keep out rain and animals.