I have encountered some issues with that recently when I finally had the courage to try out some Javascript xD I still don't get how the html is created, however there's one that is the same for every page, which you noticed already, and then in the php page I've gained the habit of echo'ing whatever new html I want to add. For example:
When I had to use jQuery on the registration page, I had to add in a new div to the HTML of the page specifically (I wanted a div to disappear and appear depending on what radio button I clicked). So like $document->add(new Comment("<div class='whatever'>")); (random form stuff I wanted to make appear or disappear here) $document->add(new Comment("</div>"));.
For input names and stuff, I checked what they were by viewing the html of the registration page and went with that
So I think you are really going to need to add in your own html and for each page specifically (echo'ing in the php file xD ) I did it through the Comment objects since that works like echos but for the inside page you are working on.
If you find another way, please share because I'm sure there are more people needing it badly xD
Also, welcome to mysidia!