Most form extensions would be sufficient for this job.
For example with RSForm (paid extension), you can build almost any kind of forms and render them as a module, using the RSForm module in any module position, or inside any article using the RSForm content plugin.
You can setup the form to send emails that you can customise to Admins, Form submission ...
This is not the direct answer to your question, it is more for information and to give you some basic direction.
It seems that you do not have full understanding of how things work, and why you can't add html in the fields there.
The fields you are trying to add html, are not designed to accept html. Furthermore, even if they would allow some html, I am not ...
By looking at updated xml file, I see only one issue that you haven't used
<fields name="params"></fields> in config tag and hence your modparams object showing blank. Try below xml snippet for configs and check one more time the $module->params values.
<fieldset name="contact_form" label="...
In the Joomla backend, go to:
Components (top menu) >> Contacts >> Contacts
Here you can add and edit contacts. There will already be a contact pre-added so you can simply open it and edit it to your liking.