The right to the ownership of a domain name

1. If the domain name that I registered had already been registered as a trademark or a company's name. Will I still have the right to the ownership of that domain name afterall?
2. After I have registered a domain name, can someone else still register that name as their trademark or company's name?

This whole area is one of the hottest topics in Intellectual Property and Internet Law.

Unfortunately, there is no easy answer. The cases go both ways, and most of the matters are negotiated out. It also depends on the amount of resources you and the other party would be willing to invest in the struggle....

For example, earlier this year the well known website eToys.com got a Federal judge to preliminarily enjoin an Internationally known artists group known as "etoy", which had registered etoy.com long before eToys.com was even formed, from using etoy.com. Worldwide protests resulted and as a PR matter, eToys.com withdrew its complaint and paid etoy's legal bills.

The real answer is that it would require litigation or arbitration to find out, based on the very specifics in your case.