Switching Primary Domains

I'm curious if I can go ahead a publish a SS site with the SS default domain to view and test it in the online environment. Once I've done this, can I then add a custom domain and maintain the integrity of the website? In other words, will it be re-published under the new domain name and with all links/functions in tact? Thanks!

Once published regardless of domain, it is published and switching domains has very little effect  if any on content. The only exception would be some internal url links.   Like a button or link to a gallery or blog post that you created on another page and added the url yourself. If you had those you would simply have to edit  the button or link and change the url to the new one if they don't work any longer.