I don't know how you have your galleries and website set up as you said your link was to a Flash Gallery and I haven't had Flash installed for years. I have a webside with a hand coded Home page that links to several Lightroom galleries and other hand coded pages with internal links back to the Home page. Each galleries photos were/are in collections and it was extremely easy to just recreate the Galleries using the new HTML5 templates once I got one to work and then upload them. I can't help you anymore than that unless you provide more detail on how you website is constructed.
You can't convert them, you'll have to create them again.
Which directory structure do you mean? The folders generated by a Flash template are different from those generated by one of the HTML5 templates. Or do you mean folders on your web server? So you put a Flash gallery for Italy pictures in www.prorszasz.com/italy/, for example, and you want to do that again. In that case, output the HTML5 and upload it to the same folder, probably after deleting whatever was in /italy/. Do you know how to access folders on your web site using Dreamweaver or some other FTP tool? But basically you're going to have to recreate, not convert.