Thank you for your answer.
The problem is that I am also considering doing other online business, and I will probably put them all under the same Ltd company. Thus, if the company is called AAA, then the first website might be called BBB.xxx the second CCC.xxx, etc.
Can anyone open a company called BBB and trade with BBB.yyy
Note the name in the domain from xxx to yyy. The name it'll be the same, but not the subdomain.
What about using the name in America? Would it be protected?