1) How is work going? Is this correct? I mean office work.
The question itself is fine. It doesn't matter whether you mean office work or not here though. If you simply ask someone "How's work going?" it's a general question simply asking if things are OK at work.
2) How are you doing? Is this correct?
This is fine if you are asking a general question about someone's health or just how things are going in their life.
3) "We are sitting in the food court." Or should I say "We are sitting at the food court"? (It's an open space.)
Student or Learner