hostgator and godaddy both have a cpanel. they don't call it that but that is what it is.
any hosting account you buy will have a hosting control panel where you create your website, create email accounts, create and work with databases, etc. etc. etc.
one of the things that most hosts offer now days is some type of website builder software.
when you finish building your website you will have a set of html, css, and possibly php files that make up all the pages of your website.
you will then be able to use your ftp program on your computer to access those files on your hosting account/server.
you can download the phpbb zip file from here, unzip it on your computer, then use your ftp program to upload the included files to your hosting account.
by the way, you absolutely will have a root directory for your account, not sure why they told you that you wouldn't.
then, after you have created a new database using your hosting account control panel, you can install phpbb by following the instructions found in the download or by following them here at phpbb.com
you will certainly be able to open the html files of your website and add whatever links you wish.
the next time you contact anyone at hostgator or godaddy you should ask for someone that knows what the hell they are talking about.
actually, you shouldn't contact either one of them at all. you should find a decent host that knows what they are doing.
hostgator used to be a good host but they sold out to some huge company that doesn't care about customer support .
godaddy is not and never was a decent host. they decided that they should setup all their servers, customer accounts etc. completely differently than all the rest of the industry and then hired idiots to man the support lines etc.
sorry for my rant.
go here: https://webhostingtalk.com
and find a good webhost.