Recently, I bought an aged domain name which is a combination of two common words (i.e. something like birdsworld.com). I intended to use it to establish a related online business. Before registration, I did a search on TESS to see if a similar trademark existed. The results showed a single record dated some years ago which had a 'dead' statue. But, after I bought the domain, I realized that there is a company with the same name and working in the same business in Texas (i.e. something like birdsworld CO). Now, should I be concerned about legal issues relating to this domain name if I use it? Thanks for your help!
Doing a TESS search is only the first step, you need to check for any existing usage. The answer to your question is - maybe. It's really going to depend on the exact name, the scope of the other usage(s) (do they only do business in Texas) etc. It's possible the usage is generic. Also, infringement does not need to be exact - phonetic, similar spellings, etc., can all be infringement. You should really talk to an intellectual property rights attorney. You really don't want to put a lot of time into a domain only to have it taken or be sued.
As mjewel said, any trademark turns on a number of facts. Even a "live" registration doesn't necessarily bar your use (that's why we can have Apple Vacations, Apple Computers and Apple-Rent-A-Car). On the other hand, the absence of a registration does not mean you are free and clear. What is called "common law" trademark rights arise simply by using the mark. If someone else is using the mark, you need to decide whether your usage infringes on their rights. That is very difficult for even an expert to evaluate and the answers are almost never black and white. If you are going to make a significant investment in time or money in your website, you should talk to an IP attorney to understand the risks. If you don't mind changing the name if you get a cease and desist letter someday (and don't plan to blatantly leach off the existing business), then the worst that is likely to happen in the real world is that you'll have to change the name and possibly give up the domain to the trademark owner. There is no guarantee, but most people who have trademarks simply want the infringer to stop and would only seek money damages in the most egregious cases.