Well you can tell based on the Scales of the Province.

If you are attacking a province that has that blue "Cold" symbol or the red/yellow "Heat" symbol there is a good chance that the weather will be like the scale.

I don't recall if Storms are random (though I could be wrong), they are caused by casting a spell /Storm/ or items Staff of Storms, Tempest.

The same can be said for /Rain/ it's a spell.

I don't think snowing causes any effects. And I have yet to see a sandstorm

Storm has quite a few effects. It lowers precision so archers are less effective. Stops flying (except for certain units and units with certain items) and increases the effectiveness of certain spells.