I would like to know if it is possible to have one public category where guests are allowed to write and to enable moderation only for this specific category ?

Yes, that is possible to allow public user to write a discussion post in one category, but unfortunately currently that was not possible to allow to set post moderation in certain category.

Is there a possibility e.g. to ask for email address or other things only for this category?

As long as that is public user trying to create a new discussion, it will always ask for user to enter email address.