Not only "can" you use your DQ on a day you don't go to the parks, you probably "should"...that way you get a whole 'nother Disney Day!
DQ is a wonderful way to spend a family day. We started out all together doing the larger ride/event things (these are most enjoyable as a family, especially the Pirate Ship and the "shoot the balls at the other cars while riding in your own" ride (I forget the name lol) There are about 6-7-8 of these larger ones, including one where you design your own roller coaster ride and then ride it...and one where you work together on a space ship to rescue the other astronauts who are on a planet and you have to pick them up while fending off aliens.
I addition to this there are many many arcade games/rides. My husband and teens especially liked the simulated Indy Race Track cars that let 4 people at a time compete against each other...and the star cycles...and the motorcycles...
You can check your email etc. in the Internet Cafe in DQ. I didn't do this but I believe it is complimentary. There are also many video/arcade game machines...including Frogger and Pac-Man. Our family never went to the one level where you can pay additional money for some games that give you tickets for prizes if you win. They were all more than content with the games/rides that came with the ticket (which is most of them).
DQ is located in Downtown Disney so it is very convenient to take a break for lunch or dinner. We went early at 8:30 (DQ opens around 10 or 11am I believe) and then as well as later in the day did some shopping, had sundaes at Ghirradelli's, played in the Leggo place (robots!...as well as regular Leggos)...we left DQ at about midnight I think...well satisfied after a full day of fun. Oh, and we ate at the Rainforest Cafe in Downtown Disney that day too.