The rain is nothing more than a graphical effect for the player, it affects nothing in the world aside from obscuring our vision.
The devs have a water overhaul planned for A21 (subject to change), and I'm not sure what that all means. Does it mean different speeds of currents? Better textures? Rain to be improved and effect the world? Water to properly level if bucketed? It could mean anything regarding water.
That would be awesome but no, they do not fill with rain water now. The bits you see with water in them were generated that way when the map was made.I haven't seen the "natural" crevices fill with water from rain, but the little drain ditches on the sides of the road do!