Yes. A domain name can be purchased from a domain registrar such as GoDaddy. Services such as DynDNS provide subdomains (for example, ) free of charge. Also, keep in mind that even once you have acquired a domain name, you will have to purchase hosting services and DNS services separately, though most registrars offer complete packages.
You do have to pay for domain names due to the large maintenance cost of servers. However, you can get free subdomain names (i.e. /yoursite) from various websites such as or .
Yes. you have to pay to register the domain name. The price can vary by the additional services that you want with it. Also some of the more common names that are. com can cost more.
If you are interested in starting your own business online, or just creating a fun site for your friends, then you must have a domain name that you pay for.
You will have to pay roughly $8-$10 a year for a domain name. Usually, you can't get a domain name for a one-time fee. Some of the companies offer a "free" domain for life if you subscribe to their hosting package (which usually goes by a monthly fee).