Will comprehensive cover the damages to your car from hitting a deer?

If you hit a deer, that's a collision and is paid out from your Collision coverage. Comprehensive is for Physical damage done to your car, such as a tree falling on it, theft, vandalism and fire.

Yes, the comprehensive portion of your auto policy covers collision with an animal, such as a deer. It would be subject to your deductible.

http://www.iii.org/individuals/auto/a/basic/

Actually it's covered out of comprehensive coverage and in most cases your rates will not go up. Comprehensive covers acts of nature including colliding with an animal