The Millenium Hotel is not really a good location and it's kind of old. Stay closer to Myeongdong or City Hall. Better for walking around to sites easily.
Gimpo is a much, much faster airport. There are literally just a few flights per day, so there are no lines for immigration and you can be on the train within 20 minutes of stepping off the plane. Incheon can vary wildly.
You don't have to stay near the airport. Limousine buses and trains start at 5 AM. I would recommend though with four people book an international taxi. intnatltaxi.co.kr They are run by government and offer flat rate and schedule pickup. With four people the per person price is similar to the bus ticket price per person. You can also take them from the airport as well.
The best place for food as a visitor would be the area near Jongno Avenue of Youth. It is a real local area for nightlife and most places have English menus. Myeongdong has English menus as well, but there is nothing local about Myeongdong.