Is cooking a survival skills?

Yes, but it's a broad concept. Very basic cooking skills, like starting a fire and cooking meat, or boiling water for drinking, absolutely, in most situations those basic skills could help you to gather and eat food from an environment safely.
"Cooking skills" can cover a lot of different areas... and the value of certain cooking skills will depend on context.
In other words, Knowing which salad dressing goes well with shrimp will not necessarily help you if you are lost in the Australian Outback.
Knowing how to prepare certain types of food can be an important survival skill, absolutely, in the right context.
For example, there are a few fish that are poisonous if not prepared in the right way. They could be an important food source, if they are available, and you happen to know how to prepare and cook them... but that knowledge is no help in the Himalayan mountains, or the Sahara desert. Different cooking skills would apply there.

Not really. Firstly, in the wild everything you can eat is raw, therefore you should focus more on how to kill an animal and eat plants rather than cooking them. By the way, humans are not actually omnivores, they are closer to herbivores, thus plants is what you should eat mostly. Secondly, even if you had to cook food, how would you be able to do so in the wild?

In the case of leaving at home alone, then yes, you should know how to cook food. But there are restaurants so it wouldn't be that much of a problem.

No, you really don't have to "make" any food. You can just eat it as it is, but it often taste better if you cook it as well.

Cooking is an important skill but when you're in survival mode the most important thing to learn is what you can eat and what you can't eat without cooking it yes I know that sounds disgusting but often when you are stuck in the wilderness by yourself you can't always build a fire sometimes you have to eat meat raw you have to know what you can eat and what you can't eat and how to minimize the bacteria if you have to eat it raw

Yes