What's the effect of cold weather on...
 

1. string tension
Does it make the stringbed tighter? So string a couple lbs lower in the winter?

2. balls
Do they travel further/shorter? Feel heavier/lighter?

How should one compensate his game, or does it not matter at all?

Strings feel more harsh, less pliant, so more dead.
Everything moves slower, feels heavier.
Compensate. Depends on your game and what you can bring to the table.