You mention copyright, but this looks to be a trademark issue.
Also, all of the terms at play here are pretty generic, or at least highly descriptive. Thus, the likelihood of either entity having strong trademark rights is pretty low. These URLs appear to have been specifically chosen for their descriptive, generic terms. Trademark rights are earned by brands that are either inherently distinctive, or have become distinctive through consistent use, promotion, or the like. Here, none of the terms seems distinctive, and thus Company A's trademark rights would be weak at best.
(949) 721-6380 - Of course there's more to it! Plus, we don't have an attorney-client relationship. This brief comment is for information only, and must not be relied upon as legal advice.
There are two schools of thought on choosing brand names / trade names. One is that the name ought to describe what the brand represents. That is intuitively obvious, but it has the downside that such descriptive names have little or no protective value in any trademark dispute.
The law instead calls for distinctive trade names (and logos). Te example that comes to my mind is Exxon [R]. What is an Exxon. Nowadays it is a brand of oil products etc. But when it was rolled out it had no meaning in any language, guaranteed. Instantly register-able at the PTO and very strong in a dispute. With domain name issues it is also very strong in an ICANN dispute.
So the second school is, do not pick a name and fall in love with it. Instead, do what the big boys do, on a smaller scale. Form a committee of the owner plus a branding expert plus a TM lawyer, and work the problem together.
Licensed in Maryland with offices in Maryland and Oregon. Information here is general, does not create a lawyer-client relationship, and is not a substitute for consulting with an experienced attorney on the specifics of your situation.
I question the capitalization of Cheap Books USA, as this indicates not that you are simply "selling cheap books in the USA" but that you are implying to visitors to your website that your company name is indeed "Cheap Books USA".
On the surface it appears as if there might be a slim arguable "likelihood of confusion" in your use of: "Welcome to "Cheap Books USA" as compared with an established company using the name Books USA. (Dot com or not at the end.)
It sounds as if you are being threatened by Company A, in which case I suggest you contact an attorney for a confidential consultation and a more informed decision could be reached.
My disclaimer is simply that Avvo already has an adequate disclaimer.