Wow, awesome itinerary. Good for you.

Of course you have enough time to go into the city for night in Bangkok and Istanbul.

In Istanbul, I would recommend a hotel in the old city, or even closer to the airport like the Sheraton Atakoy or Courtyard. A visa costs about $20. You might want to check out the nightlife in the Taksim area across the Golden Horn, though your best bet would still be a hotel on the old city side so you don't have to deal with carrying luggage if you check out of a hotel on the new side in the morning and go to the old city for sightseeing.

In Singapore, you can take the train into town and walk around, perhaps have lunch at a hawker center. They have a left-luggage area at the airport (as does IST) which could make your life easier.

Cairo will be a bit more complicated. The only option is taxis both ways, which will be at times of heavy traffic, though if you're up for the adventure if beats spending 14 hours in the airport. If you want to spring for a hotel, Le Meridien Heliopolis or the CityStars complex are within striking distance of the airport.