Kids here do pretty serious parkas, usually with hoods. Many will wear hats too but a hood is something that can be quickly flipped up when thee's a surprise rain or snow storm, and for some girls, it's preferable because it doesn't squash their hair as much as hats. Others just wear hats because they don't want to freeze and they don't get sucked into the "looks" pressure. I have an LL Bean parka which wasn't cheap but it's incredibly strong and lasts for years. That might not work so well with a growing child who's fussy though.
I have 2 coats with zip out linings, and both have a button in the sleeve of the main coat and a loop on the wrist of the lining, so they hook just fine and never get messed up. One is a rain jacket (hip length) and the other is a trench coat. You could always add that to any jacket where the lining is slipping - go to any sewing store and tell the clerk what you need help with, and there will be an assortment of notions you can choose from to just hand sew in place.
We have a lot of resale shops - I don't know why her dad is so anti-repurposing! Especially when many of the shops support charities. I don't know if you have Burlington Coat Factory stores there - it's a chain in the northeast but I don't know how far they reach. They have a huge selection if you're willing to spend time looking. Check your newspapers for coupons in the inserts - sometimes those help with a big purchase too. I usually do well with Marshall's, not so much with TJ Maxx even though they are owned by the same company.