Can I build a car-port on the side of my garage?
This is highly dubious since you mention snow. The current garage sidewall is not high enough to support the high side of a shed roof over your proposed additional space, and the additional roof (if you just slap it on there anyway) will introduce additional snow-loading to the garage roof that it's probably not designed for. You can get some massive snow drifts in that configuration. 

You would do better to build a totally separate structure away from the current garage. Once again, given the mention of snow, that structure would probably better be a garage - aside from people blindly following trends from elsewhere-that-it-does-not-snow-in, carports have never been all that popular in snow country because the car still gets snow on, around, and under it anytime the wind blows, so it's barely better than parking it out in the open.