Ad not displaying.
In your Admin CP there should be some options for your install's path (ex: public_html/foldername/) or you may have an option selected that lets phpBB automatically detect its own path. There should also be an option to change the cookie settings to use .NEW-DOMAIN.com as the settings instead of .old-domain.com! Make sure those are altered. In addition, your forum's site/board URL setting may need to be updated to link correctly, but I forget if phpBB does that automatically based on your other settings. It might.
There should be some sort of server configuration tab/link in your ACP where you can view things like script path, cookie settings, etc.
Once you update those settings you *should* be able to just move your files/database to the new domain. You can use phpMyAdmin export function to download your .SQL file & then re-upload in the new domain's database manager. Or, depending on your setup you may be able to keep the database there and just connect to it as you already have been doing. It really depends on where your old/new domains are in relation to your server & permission settings.
You should to modify .htaccess on your old domain so that all URLs are PERMANENTLY REDIRECTED to the new domain. This will help you preserve SEO / rankings for your links, and make sure any links scattered around the web end up in the right place. If you do not do this then visitors will see 404 NOT FOUND when they click an old link. Definitely redirect them with a Google-complying method (aka permanent redirects, not in-line HTML redirects or redirecting every page to the homepage of your new forum)! You will need to use some sort of regex rule to capture the path (ex: old-site.com/path/to/page) and redirect to the same page on new domain (ex: new-site.com/path/to/page).
Ad not displaying.