hi,
I think it's okay to take him out for just a few min. we got our puppy in Dec. & he is 14 weeks now, we live in Ohio(also very cold) & we just keep taking him out. we have a fenced in patio, so he is safe to go out on his own, but we watch him the entire time. we give him lots of praise & treats every time he potty's & he gets it, he comes in the house & sits & waits for his treats. we are doing crate training, but he won't dirty his crate, so that helps us. we have a pee pad in the house, but he ignores it. just keep being consistent, he sounds really smart, we have only taught Lucky sit & were working on paw
good luck