My wife has a cold. Can I catch it by sharing a toothbrush with her, even if we dip the toothbrush in mouthwash between uses?

Sterilization. Mouthwash will not sterilize the toothbrush. They're not that expensive to have your own. You're most likely going to catch her cold through other means anyway.
Yes. Mouthwashes do not necessarily kill all the bacteria. Also, when you say you dip the brush I assume it's just the end with the bristles. The handle of the brush was touched by your wife and most likely has the cold germs on it, too. Why don't you simply buy a brush for yourself? They're not that expensive!
No. Airborne disease. Don't shake her hands after she sneezes but if you've been sharing the same "air space", you will likely get it.