I really don't see the point of a contact for on a forum. Mine usually don't have one. That being said, I use ProBoards and they allow the option for private boards meaning that only the user who creates a thread and specified staff members can see the posts. It's quite appropriate and hence why I don't use contact forms.
I use phpbb so have the inbuilt one that sends emails to the admin email account. It's not used much, but it's useful for members who can't log in and need help.
Yes, always have one.
I find it is one of the most important part of the website.
If a user/client is having problem registering and he has no information or a place for him to contact you on the issue, it could lead to big problems.
I have never really seen a use for a contact form, usually I just have my email out there for everyone and if they need to get a hold of me they can simply email me. It's much easier than putting in a contact form. 😛
Always had contact forms on my forums, just because I liked to give people a hint of help if they were having problems registering or logging in. Saying that though, I can't remember receiving more than a couple submissions... most of the stuff submitted through my contact forms was usually spam.