FormLayoutFunction will do some of the things you want. For example, here I set the input labels to be White.

f = FormFunction[{"name" -> "String", "question" -> "TextArea"}, 
  Grid[{{"your name is"}, {#name}, {"your question is"}, {#question}},
     Alignment -> {Center, Baseline}] &, 
  FormLayoutFunction -> 
   Function[
    Grid[{{Style[#["name", "Label"], White], Style[#["name", "Control"],  White]}, 
         {Style[#["question", "Label"], White], Style[#["question", "Control"], White]}}, Alignment -> Left]]]

I don't know how to "hide" the boxes or "simplify" the "Powered by Wolfram". The wolfram license agreement on the page you linked states: 1. You will not remove, obscure, or otherwise disable the footer attribution information and links on Wolfram-produced forms.

So maybe you shouldn't do this!

As for where to put the html, I'm not sure what you mean. The simple answer is, in a web page. When you view the web page in a browser, the form will appear basically the same as if you visit it's url directly.