The question of is (whatever) a trademark, is really the following question:
"Is (whatever) a distinctive term applied to goods or services by a particular source of those goods and services in order to distinguish them from others in the same market?"
Or maybe I can put this in simpler terms.
I have a string of letters in a domain name. Let's say it's "farzqle".
I want to know if someone sells a product or service branded as "farzqle".
What would be the simplest way to do that?
Seriously. How much thinking does that require?
Suggesting that the first-pass answer to that question is to go to some database of trademark registrations somewhere (and believing that some random person is going to be able to actually interpret and understand the data from that database), requires a real dedication to missing the entire point.