Can I edit/add fields of Contact Form at Emmet Front Page?

Yes, there is option to add third party form using shortcode at Pro version of Emmet Theme. Thus you can install third party plugins like Contact Form 7 or our Simple Contact Form addon (addon will work if you use Content Editor plugin). There is huge choice of free plugins to add Contact Form. Then create your form using this plugin, take its shortcode and paste to Appearance>Customize>Contact Us section>Contact form shortcode(replace built-in) box. Then you will see your custom form instead of default one.