Cold air is harder to inhale too, once I get going in the cold I'm fine... you're likely to pick up injuries if you haven't warmed up properly first though...

If your running hard in the could then you WILL get injuries... gotta warm up (get the blood pumping to your muscles... when you're not using them less blood is going there... when it's cold even less because your body wants to try and keep the heat in) properly first AND stretch because your tendons and muscles will also be tighter and shorter in the cold.

I wouldn't be too bothered about your core temperature... once you start training your body is generally trying to keep it down because of the heat that exercise generates... but yeah... if you're in really cold conditions even with the heat that exercise generates your body may have to actually try and raise your temperature... which is gonna play hell with your muscles... we're talking up a mountain or something here though... :P

The main thing to do is not to cool down out doors... not even when you've finished training... some times in the summer I'll cool down outside and wak the last 1/5th of a mile of my run... can't in the winter... it's bad at the end of your session and it's suicide in the middle of it... It's almost impossible to get warmed up properly in the cold... you'll have to stop your run or what ever... you'll tighten up and tear something.

That's why Football (Soccer :P) players get injured in the winter all the time... a lot of them allow themselves to cool down during the game on the cold pitch then suddenly sprint a ball or something.