Assigning only certain categories to a custom post

Is there a way to assign certain categories to show themselves in a custom post?

For instance, I have the traditional post “Post” and just created the custom post “Games.”

I have the categories News, Articles, Xbox, PS3, and PC.

When I click on my Games custom post to create a new game, I only want Xbox, PS3, and PC to show up… but not News or Articles.

Similarly, for my default Post, I only want News and Articles to show up.

Is this possible?

As a side note: is it possible to have certain categories automatically checked when I begin a post? Like if I want “News” to be checked by default, and if I don’t want it to be News, then I can manually uncheck it?

Thanks all!