Are there any regulations of one owning a domain registration company? What are the qualifications if any are there? Are there any fees to be paid?

Yes, there are fees. Unfortunately I only know Auda (which is the Australian .com.au registra) they charge about $2,000 - $3,000 a year for the service and get you to charge above $30's a year.
So I am sure they will be one for .com's or .net's out there. Just google 'domain registrar'