So I'm using Host Excellence and they have a way to install the forum within the current control panel... Just by enabling the PhpBB... Do I still need to "upload" the latest forum files from another website? Or does HE just do that when in the control panel?
You wouldn't have to upload all the files but I wouldn't recommend you to install it that way. Thing is HostExcellence doesn't always have the newest version installed in that one-click install. When PhpBB Forum version 3 appeared they still had version 2 as one-click installer. I don't know whether they provide version 3 now or not. Anyway I would really recommend you to avoid the one-click install and install it the traditional way by uploading the needed files yourself and go through the install process as described herehttp://phpbbfaq.net/site/PhpBB_Hosting_FTP_HE.html
Same for any other hosting: they don't always provide the newest version for one-click install.
In the meantime, I would like to imbed the forum into one of my pages, just to keep my theme... Would that work using iFrame?
Yes it works putting it in an iFrame.
How can I keep it secure from the outside world? I only want MY members to have access to the forum and no one else. I don't even want others to be able to read or access it.
Control Panel=>General=>User registration settings=>Account activation: Disable, this disables new registrations completely.
How you would create new users I don't know. You would probably have to change Account activation: Disable to None, then go through the normal registration process and change it back to Disable. I don't think there is a way to add new user in the Administration Control Panel itself.
Also you would have to set the user permissions of Guests and Bots so they wouldn't be allowed to Read (hence No Access)http://phpbbfaq.net/site/PhpBB_Set_up.html
You could also password protect the /forum/ folder to increase securityhttp://iwebfaq.org/site/iWeb_Folder_FTP_HE.html#6