How can I use custom domain for my forms?

If we sign up for paid service, will the url of the link where a person completes a  form always show Jotform?

You should use the custom domain feature to hide our name in the URLs. And the custom domain feature is only available with JotForm Enterprise. 
You can change the form ID part of the URL as a free account. 
Guide: How-to-Create-a-Custom-URL-of-a-Form