Do animals and birds grow warmer coats in colder winters?

Birds don't.

Some do, like wolves and foxes. The gray wolf has very dense and fluffy winter fur, with short underfur and long, coarse guard hairs. Most of the underfur and some of the guard hairs are shed in the spring and grow back in the autumn period.

Birds don't.

Mammals often do that. Most other animals don't do that because they don't generate internal body heat. Birds do molt their feathers, but they don't have a thicker winter coat. Instead, they often migrate away from cold areas if they cannot stand the cold or if they need to go to a place with food. Birds can easily fly hundreds of miles in a short time, but mammals cannot do that so they need to adapt one way or another.

Yes. Many animals can strengthen their coat to prepare for winter, or shed their fur to reduce summer's heat.

It is actually true.. they develop fats in their body covered by fur which helps in proper insulation...