I do have two questions:

I have a paid web-hosting and a domain-name. (www.caselawofeu.com)

How is it possible to transfer from present server to a server of a new provider. And how does the transfer of domain-name work? 

Can I pay the server in a new provider and to move the site now? And after that, domai-name will be allocated to the new server provider? Still I have payed my hosting. 

Is there documentation, how to do that?

I newer done this before. Thanks for all of your answers.