Stationary bike, elliptical, rowing, astanga yoga... anything that gets your heart rate up. Cardio is the only way to rid yourself of that belly fat. You can strenthen and tone the mucles underneath, but the fat will remain.
Also, I find that I can run if I wait until 2 hours after I eat.
Can you not do weight training at all, or just not exercises when you lay down?
I haven't been pregnant so I can't answer your other question.