You seem to be confusing corporate names and trade names with trademarks. Corporate filings are made with the secretary of state for the state of incorporation. While you may file for a state trademark registration, you probably would be better off with a federal registration from the USPTO. You need to do some more homework and then sit down with a trademark attorney to help clear up your misconceptions about trademark law.
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Q: "Can ABC LLC. create multiple websites called facebook, myspace and yelp and receive trademark rights without filing for them nor including them as DBAs?"
Answer: Yes. Trademark rights are based on use, so if used on multiple sites that would likely yield trademark (you perhaps mean servicemark rather than trademark) rights. Whether or not fictitious name filing (dba) is made would not determine whether ABC LLC has trademark or servicemark rights. The dba filing is for purposes of complying with notice requirements so people know who to notify if they have a complaint about the business.
Q: "Is a website and its name a company?"
A: Not necessarily. A website could owned by an individual in Tonga or Tuvalu Islands or Zimbabwe or Paris or Chicago, and it might be anonymized so you cannot find out.
Q: "I understand a company like an LLC creates a website but what does that make the website? "
A: Huh? Are you asking if a domain name is a trademark? Maybe. Depends on how it's used. It's really an address, but trademark owners tend to launch websites having their trademarks as domain names. If you are asking something else, please rephrase and repost.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
This is your second question about trademark law. You need to read a few books that explain trademarks [and other types of intellectual property] to business folks. See the link below.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
This is really about the difference between a corporate name which can operate businesses under separate trade names or divisions which may require filing a d/b/a registration and/or trademark registrations. The easiest way to think of it is General Motors which has lots of different divisions and trade names and trademarks but is all one company (for now).