If I'm riding on the road, and there is a red light but a walk signal in my direction, should I go? I frequently come to stop lights in which there is an all-way stop and a pedestrian signal. I always stop at such lights, but I'm wondering if I should then bike slowly across after determining that it is safe and I will not be interfering with any pedestrians who are crossing, or if I should wait until the light turns green just as I would were I driving a car.
Only cross if you are walking and pushing your bike.  If you are riding, you are a vehicle and generally you are required to obey all traffic control devices in the same way any other vehicle operator is. There are exceptions, like the Idaho Yield laws, but generally you have to behave like a car.

The other thing to think about is that the more you behave like a vehicle, the more predictable you are, and the more predictable the behavior of other vehicle operators will be.