This question I can actually help with! One of our party is on insulin, so I have confirmed that the fridge for her room is free of charge (for medical reasons). Our TA did advise having a doctor's note confirming that she actually needed the insulin, just in case the Front Desk asks for "proof". Yes, let the hotel know before you get there - them knowing beforehand may make it easier to get the fridge when you get there.
Sorry, I can't answer your second question, but when we find out, I'll let you know. My MIL and I were discussing exactly the same question yesterday (the insulin is for her sister (our aunt)) and she was going to call the doctor's office to ask. I will be checking the airline policy as well - they may specify certain medicines that they will refrigerate for you on-board or allow you to keep in a cooler kwim?
OK, I just called the airline. If you're travelling with AlaskaAir, this is their policy:
As long as each individual vial of medicine (in this case, insulin) is clearly labelled with doctor's name, patient's name, etc., you can bring it onboard in a cooler bag with freezer packs to keep it cold. Also, again as long as each vial is clearly labelled as above, the cooler will NOT count as carryon luggage!