Is vinyl (linoleum) flooring a good idea for the bottom? I've read some people say it can make the chickens sick if they peck at it but if its completely covered by bedding they shouldn't have access.
No matter what you propose to do on practically any subject on here someone will tell you that there is something wrong with it, either potentially or definitely. They can always find something that might go wrong. To me, if your vinyl floor is waterproof and will keep the wood under it dry it will work to greatly extend the life of your coop floor. A lot of people use vinyl over wood without issues.
Can I put pine shavings over the sand in the winter to provide a warmer bedding material?
Some people use sand and hate it. Some people use sand and love it. Even if I knew a lot about your coop, your flock, your climate, and your management techniques I would not be able to tell which camp you will fall in. If you try sand and do not like it, I don't know how hard it would be for you to remove it and replace it with something else.
In winter and in summer my chickens sleep on the roosts. I don't consider a warm bedding material to be a high priority. It is generally colder at night than during the day. Why is a warm bedding material important to mature chickens?
Whether winter or summer, my concern with bedding material is that it stays dry. If it gets wet and stays wet it can become unhealthy and stink. Wet could come from your waterer, the environment such as rain or snow, or the poop building up so thick it can't dry out. That's my concern with bedding. I hardly ever change mine out, I don't need to. But if you do change it out often, how easy will that be?
My coop floor is dirt with wood shavings. It stays dry. I use droppings boards to keep the poop from building up very thick. I have weather they can stay outside all day so they are not pooping in the coop during the day and the droppings boards catch about everything at night. I should clean the coop out every fall to put that stuff on the garden so it can decompose over the winter in time for spring planting but I'm lazy. I generally clean it out every two or three years, not because I have to but because I want that stuff on my garden. Some people do deep cleaning weekly or monthly. That means change out the bedding.
I have no idea how often you will be changing out the bedding. Trial and error is probably the best method to determine that. If it starts to smell you need to change it. If you change it regularly sand is probably not the best material to use.
I have no idea what the best bedding material will be for you but I want something easy to work with, relatively inexpensive, and readily available.
Good luck and welcome to the forum!