As others have mentioned, this would be very fact specific.
"I've been told that I have a right to do business in my city using this business name, but i can't open more locations outside of my city because of their trademark. I don't have a problem with that at all. I've also been informed that since I registered my DBA during the 3 year gap in their trademark registration, my competitor does not have the right to establish locations in my town because I registered my DBA with that name during that 3 year gap period there was no trademark."
First, I would say that regardless of the 3-year gap, the other party has a duty to police their trademak rights. Courts may give 2, 3 or even 4 years for the other party to police (stop a confusingly similar mark from being used), but beyond that, you might have a strong argument for continued use of the mark in your geographic area regardless of a Federal Registration, so long as your mark was adopted in good faith. How broad a geographic area a mark is accorded could strongly depend on the industry that you are in- restaurants and banks may be given a smaller area of protection, whereas more specialized businesses may be accorded a larger area to exclude businesses with confusingly similar names. Now the flip side of preventing the other business from expanding into your geographic area would, among other things, depend on how long they have been doing buiness in your area,the industry, etc...
As also mentioned, see a qualified Trademark Attorney tha could answer questins more directed to your specific situation; not just any attorney, but one experienced in Trademark matters. Good luck.