Yes. Trademarks generally fall into two categories, strong and weak. Strong marks receive more protection than weak marks. The relative strength of a trademark depends on the distinctiveness of the name, logo or symbol being used, but weak trademarks have the potential to become stronger over time.
Business names by themselves are actually referred to as "trade names". A trade name is not the same thing as a trade mark, and many companies use their trade name on items such as letterhead, invoices and other important documents. A trademark on the other hand, is affixed to a product or a service that the business offers.
Obviously the two can overlap, and many businesses use their name both on letterhead and affix their name to products that they sell. For example Xerox is both a company name, used on letterhead, but is also branded onto their copiers and business machines.
Yes. To be clear, however, although a website or domain name may be given in full as something like http://www.companyname.com, neither the http, www or .com are part of a name that can be trademarked. Rather, it is the name in between the www and the .com that can be trademarked. A typical example of this would be a site like www.google.com , where google would be the website or domain name that is also a trademark.
Trademarks receive protection under both federal and state law. The group of federal laws that govern trademarks are collectively known as the Lanham Act (15 U.S.C. sections 1051-1127). In addition, trademarks are registered and protected by the USPTO, which is a federal entity. All states, however, have statutes that also govern the use of trade and service marks within the state.
In addition to these statutes, the U.S. legal system uses a system of "common law" based on long-standing court decisions to help decide who is entitled to use a trademark. In general, common law gives the user who first uses the trademark ownership of that trademark.