It's not the name itself that's trademarked. Trademarks are designations for goods or services, in particular classes of goods or services - please see the link below. You need to know what TM CLASSES this other LuckyPaws already has a TM in -- class 31, foodstuffs for animals? for class 25, pet clothing? for class 16, paper goods, like pet calendars and magazine?, and then you have to examine the IDENTIFICATIONS, which are the descriptions of the goods in those classes. Then you have to analyze whether this other company is likely to expand into other classes of goods and would oppose any attempt by anyone to use that TM in any class that sells pet products, and whether you yourself would want to.
TMs aren't do it yourself projects, and Avvo is littered with those who try but clearly don't understand what TMs are, how to use them, register them and maintain them. Choosing a TM is the most important business decision a business has, so hire a lawyer for help.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Have you searched Google for "LuckyPaws"? A billion folks are already using the phrase [as two words and one] to brand lots of different pet-related products and services. ALL of these folks -- just by using the phrase to brand their particular products or services -- have "trademarked" the phrase.
I think you believe that because someone has applied to register "LuckyPaws" with the federal Trademark Office to brand pet treats that person has rights that others who are already using the phrase do not. That is not true. Registration of trademark rights is just that -- registration. Trademark rights come from using the mark in commerce. Once those rights are created they can then be registered.
You need to speak with a trademark attorney to discuss the process of branding your magazine. There's much more too it than you seem to think -- and much more riding on the outcome than you likely know.
As for your specific question: is a magazine for pets and pet treats "related" goods under trademark law? Questions such as this must, in practice, be resolved based on the evidence presented. If one side or the other makes a poor evidentiary showing even a position that should prevail will not. Based on the little I know about pet products I think the weight of well-presented evidence will show the two products are related. But maybe not. There is no "legal" answer -- only a factual dispute. In other contexts, I can readily see where an industry-specific magazine and product are NOT related [car parts, for example].