I'm not sure about what it does when the horse is frozen, but you pay based on whether you choose DIY/Part/Full. Every stable charges different amounts because it's set by the owner. I always choose Full because they feed, water, and groom automatically. Saves a whole lot of time! I look for any stable that charges under $100 for Full. Usually around $50/day. Most horses can make way more than that per day if you are entering them in shows. 

I make $60000/day in halter show winnings, and $20000/day in discipline show winnings. And I pay about $1000/day in boarding fees. So it's definitely worthwhile if you've got horses who can perform

Yes even when frozen or not active you will still have to pay for boarding and some facility's will kick your horses out if they are frozen for a prolonged period of time.