Welcome to Sitepoint, @BBA;
This is the right place to get answers to your questions and guidance as you grow and learn.
It sounds as though you are interested in continuing as a Web Designer, but perhaps you were just "thrown in the deep end of the pool" on this one site. The advice you get here will differ based on whether this is a one-time project or you plan to continue (grow and learn) with Web Development.
You actually have a choice. If this is something you do not plan to do again, don't get involved in trying to manage a hosting account. Wordpress, for example, includes the hosting (go to Wordpress.com).
If your client wants a vanity domain name ("MyCompany.com") you can register the domain name and redirect it to the nnnnnn.wordpress.com subdomain that will be generated.
In the other instance where you obtain shared hosting, you are correct here. Dreamweaver should have created an entire site and it must be uploaded with respect for the folder structure. However, this does not inherently allow for the client to easily make changes to the site.
Traditionally, someone would modify the files (maybe using Dreamweaver again. But it could be a matter of editing the HTML files), and then upload the changed files to the hosting account.
You don't need any database to support a 'static' site - as you have described it.
Once again, I welcome you to this thriving community of diligent and experienced Web Designers/Developers/Marketers/Professionals and encourage you to continue asking specific questions.