Hi,
Forms created in LC Designer can be deployed as a HTML form, but ONLY if you have the LC Forms server component. Otherwise you can only really deploy as a PDF.
First off the form is saved as a Dynamic XML Form. This is okay if the form requires dynamic features, such as showing/hiding fields. But if it does not require dynamic features, then you should save it as a Static PDF Form in the save-as dialog. This is much more likely to open in the browser than a Dynamic XML Form.
The second part about filling the form and not leaving the website is not really possible. When the user clicks on the link for the Static PDF form, it should open, but it will take the full screen for Acrobat/Reader.
In relation to the email, you are using an email submission button. You would have more control over a regular button. See examples here: http://assure.ly/fiTm2i.
Generally, I would try and get a grid going and align the objects. Also you should number pages, subforms and objects as you go.
Hope that helps,
Niall