If SE indexing is not an issue, the simplest solution to #1 is to have the forum within a frame or iframe whose parent has the site wide navigation header.
Regarding #3 it is generally quite easy to edit the forum header file and remove the links to unwanted features.
#2 is more complicated but it would be a matter of your login include the process of logging in to the forum and logging out at reverse. This would involve hacking the forum login form. Your registration process should populate the appropriate fields in the forum database.
Disclaimer: Any advice given is my personal novice opinion and please follow any of my recommendations at your own risk.