I have visited the very same coffeehouse couple years ago as most of the visitors of Gaziantep. Tahmis is a famous place as it was operated as a coffeehouse since 1903 by the members of the same family. During the Ottoman period, the building was known to built to supply income to the nearby lodge used by Mevlevi dervishes.
The first coffeehouses started to open in İstanbul in the 15th century. Along way to Starbucks... And since then they have played an important role in the Turkish culture. With the effect of the Islam while women entertain themselves inside the houses, men usually meet at a coffeehouse for recreation and communication. Traditionally it is a male dominated place.
A traditional coffeehouse is very much look like Tahmis, very basic furniture and a small kitchen for tea or coffee. While customers mostly chat, it is also very common to play backgammon and other card games. 
While the modern coffee shops and chains invaded the big streets of the cities, you could still find tradional coffeehouses everywhere. In big cities just look at the side streets or lower and middle income neighbourhoods. In rural areas, they are usually at the center of the town. Although it is a male dominated place, there is no law against women to enter other than the cultural values.
I usually like to spend some time at a coffeehouse, when we are visiting small villages. It is an excellent way to meet with the locals and learn all the gossip. Why coffeehouses are still very popular? Maybe you could find the answer in an old Turkish proverb; '' one cup of coffee is worth 40 years of friendship''
