Continuity mistake: When Martin is at the Hippo Club and goes to the bathroom, he washes his hands and throws the paper on the floor. You see the paper on the floor as he leaves in a wide shot. In the last part of the scene there is another wide shot and the paper is gone.
Factual error: In the scene at the Cuppa Joe, it's a span of 3 minutes and 5 seconds from when the server takes their order to when she's walking back to the table with the food - that's not enough time for an order of pancakes to even finish cooking, let alone for all of those ingredients to be cooked and plated.
Answer: No, Doom II was never available in a commerical arcade cabinet. It must have been specifically designed for the movie.
Ral0618