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.
I'm pretty sure it's illegal. Of course, different areas might have different laws on the matter, but the guideline is "if you're riding your bike, you're a vehicle and should obey all the same rules". I suppose if you wanted to cross on the crosswalk, you could dismount, and walk your bike across, but I doubt that's worth the effort.
I don't pay attention to walk signals, but if I know the way a particular light's cycle works, I will indeed go through when it's red.
In California we have a lot of protected left turns, where the folks turning left get a green arrow and the opposing straight-through traffic still has a red light. Protected lefts generally happen with two opposing left turn lanes simultaneously getting the arrow. Often, the turn lanes will empty before the green arrow goes away. If the turn lane across the intersection from you still has an arrow, but the lane itself is devoid of cars, then you can safely cross the intersection.
In order to reduce the ire from automobile drivers, I refrain from blatantly crossing red intersections, but instead use the above knowledge to get a head start on cars before the light changes so that I can be through the intersection before they start moving.
It is, as usual, advisable to learn your local regulations. For example, Washington state law explicitly allows cyclists to ride on sidewalks and in crosswalks, as long as they yield to pedestrians while doing so. In your given scenario, use due caution, motorists noticing a red light on the cross street may not expect someone in the crosswalk in front of them.