Add contact form?

The login and register buttons look good in the popups.

Is there a way to show a contact form in the same way?

Sorry but this template are prepared to show only two popups. You can load contact form on login position instead of login form.

Could you recommend a contact form to position in place of the login form?

There are many of them on JED.
Personally I am using this one: http://extensions.joomla.org/extensions ... forms/7901