I'm thinking of using AdSense as my ad network and I'm wondering if I should add the forum in their site section, where they ask you to link the site, so they can check if its applicable to run ads on that site.

Is that necessary? Or should I just create an ad unit and paste the code in the footer, etc...?

Have you asked them about it?

Hello before you can add your own ad's you must pay for a package and remove the ad's of our's for everyone.

Once you have done that you must add your forum address in the site section of the new host.
if you don't add it then it will not find your site and it will not ok your ad's