HI there!
I live in Northern NY - and boy does your story bring back memories! We got Tucker last January - when it was below zero, extremely windy, and snowed for WEEKS straight. Tucker learned about house training quite fast - which is why he ran and hid every time he had to pee or poop! He didn't want to be taken out in the cold. And when we did catch him fast enough and took him outside? He'd stand there and just YOWL pathetically and keep hold up his paws trying to get me to pick him up.
Finally, I gave up on the battle and just bought piddle pads. I put them down next to the door we wanted him to go out and let him use those. I know some people say that it confuses the dog, but it didn't Tucker at all. As soon as it got warmer, I took a partially used piddle pad out and laid it in the snow right where I wanted him to go. I also used bell training and he picked right up on it.
Good Luck!