If they fork the project, will it be public or private? They do not have a paid GitHub account.
No, forking a private repository keeps it private, even if the person who forked it has a free account and cannot make private repositories. We do this at work - the company has a paid account, and each developer uses their own (sometimes free) account.
Is there any way to allow access to issues tracker for the private project so that it is available to public?
Not that I'm aware of. It is the repository that is public/private, not specific elements of it. GitHub themselves recommend creating a second repository for the issue tracker.