The rain can fally gently but still in a large quantity (thick) or it can pelt down from the sky and be described as a driving rain. The description has to do with how hard and fast it is hitting the ground whereas thickness would have to do with the number of drops falling in a given area in a given amount of time, whatever their speed or force.

Wouldn't smoking just mean that there was smoke from interior fires coming from the buildings? It is January, so presumably all buildings needed to be heated in some way. Maybe the smoke shows up more in the rain.