Help: Cold Engine...car shakes

Hey guyz, if anybody could help. I have 97 Corolla and recently keeps vibrating when I start it up in freezing morning (the temperature is probably less than -10 oF). It lasts commonly 15~20 minutes and back to normal. If I try to drive in this period, there is less power and the car shakes. Even bad, if I press gas a little more, check engine light will turn on. However, if I let the car idle there with 15 minutes also to warm up, everything looks fine.

Any suggestions.. Thank you!

Loose timing belt.