To make the compound noun 'xx club', you need a gerund or a noun as the first word. For example, 'singing club' or 'music club'. 'Cooking club' is fine, or in British English you can also say 'cookery club'.

It would be a cooking class if you were learning how to cook. If everybody already knew and you got together to share recipes, cook together and share the food, then it could be called a cooking club or a cooking group.

Cooking club is correct. A Cook club would mean a club for people whose surname is Cook.

Yes. Always use the gerund form to describe gatherings where people do hobbies where the hobby is a verb.* Cooking club, singing club (although as the above poster says, I can't say I have ever heard of one), reading club/group, hiking, fencing, cycling, bird-watching, etc. etc. * Sometimes the hobby might be a noun and in that case you would, obviously, just use the noun: Chess club, sports club, and so on.

Glee club: The term remains in contemporary use, however, for choirs established in North American colleges, universities, and high schools, although most American glee clubs are choruses in the standard sense, and rarely perform glees. from https://en.wikipedia.org/wiki/Glee_club