If I take my cat from a warm climate and move him to a colder climate will his winter coat get more thick?
Not immediately, no. The coat grows in gradually (but sheds rapidly!). It's too late now for your cat to grow a winter coat, you will need to keep him indoors for this winter if you move now.
Yes but they need a full season to build up that double coat. If you put him outside without it, he will freeze to death. Starting in late August, my cat eats more, to gain fat and a double coat for winter. But then in April, starts to shed. I have to take her to a groomer in May to get her shaved, otherwise cat hair gets everywhere and she chokes on the fur trying to groom herself. She refuses to let me brush her...grooming a cat is $25 and worth every penny.
Many animals, regardless of what climate they are in automatically grow and shed both summer and winter coats. Maybe it will get a bit thicker, maybe not. Now, I can't speak for all cats and owners, but when I moved from California to Colorado, I noticed that my long-haired cat had a lot more fur around his paws than ever before. We call him "Tufty-Paws" now.