You will need a hosting with your free domain. Check if you have been provided with free hosting. You can use lot of programs to design your website like dream viewer or Microsoft front page. Easiest...
go wix.com
Yes
Your best bet for a 100% free domain that you have control over the DNS is either win it in a contest somewhere
I don't know of any site that offers both a free domain and free hosting to be honest, you'll probably need separate sources for them both.