Pigeons - I guess I didn't pay close enough attention. I don't think pigeons would be TOO hard to find, but they are typically more urban - I think they like art galleries and things like that.
I think if you find pigeons people would be happy to let youshoot them.
Chukar, according to the range map at whatbird.com says they are no where NEAR MN as far as natives are concerned. That said, If you have access to land near a shooting preserve and permission to shoot it, you COULD shoot a Chukar if it was on your land.
Same thing on quail really - a bobwhites range is very limited into MN, occupying the Mississippi river valley. As you mentioned though, thats out of the question. Anything else would only occur as an escapee of some breeding program or shooting preserve.
It seems like an awful lot of work for a hit or miss (mostly Miss) shooting opportunity.