Who out there is offering a fully automated hosting solution? Are customers able to go to your website, check for domain availability, register, etc..? If so, what software do you use? What are the costs?

 

Hm, your question is not quite clear.
Plesk clients are able to check how their domains work even not being registered. I can be dome using Plesk feature called 'sitepreview'.
To register domain you can contact any registar which you like. You can check domain registration for example via http://dnsstuff.com

 

You could try something like modernbill to automate signups. It starts at $25/m or $175 for 250 clients. Then you'd need to register with a wholesale domain company like enom or modernbills company. This typically is about $100 up front in prepurchased domains.