The medium weight is what I have for all of mine and they stay plenty warm. I am in southern Middle Tennessee, so a medium weight should be enough for the cold weather you have.
I wouldn't leave it on her when she's in her stall; if she's still shivering in her stall, something's wrong and I would get the vet out to look at her.
When she's ready for turnout, you will have to use your best judgement, whether to blanket or not --- she needs the sunshine and Vitamin D.
If the weather is raw (windy, no sun, with or without rain), I would blanket her if the temps are under the mid-40's.
On the other hand, if the sun is out, the air is dry, there's no wind and the temps are down in the 30's, I might turn her out without the blanket and watch her.
Of course, when the sun goes down this time of year, things chill down pretty quick and she needs to get in the barn if she's without a blanket.
I don't believe in blanketing a horse but I have no choice with my 17 yr old. What I am trying to say is to use your best judgement and try not to get her used to a blanket every day.
While diet helps, it's not the be-all-end-all for a situation like this, when the horse just doesn't want to grow a winter coat. My 17 yr old was born and lived 100 miles NE of me for 11 of his 17 years. That's the KY/TN border - a lot colder than down here on the TN/AL border and he's never had any sort of thick coat. He barely gets any leg hair and the hair on his coronet bands rarely needs clipped
If your horse is not insulin resistant, you might try adding some crimped or clipped oats to her diet. NOT corn because it's harder to digest than oats. Oats are somewhat of a warming food.
Make sure the folks feeding her give her plenty of hay; if they don't co-operate you can also add some timothy pellets to her diet. One dry pound of timothy pellets equals one pound of hay.
She should be receiving 1.5% to 2% of her body weight in forage every day. More in the winter as far as I'm concerned but that's my opinion
Good luck keeping her warm and chubby this winter