The stationary bike nearly immobilizes the Achilles Tendon, which wraps around the Calcaneus bone. This is protective. Swimming may not be as safe in your case, since you would probably be walking out to the pool in bare feet, exposing the Heel (Calcaneus) bone to further injury.

Moderate increases in Frequency, Mileage, or Speed are OK by themselves, but not all together. A radical increase in any, or especially all of these is asking for trouble. The patience to allow your body to grow into what you want it to be is the most important skill you can acquire.

Biking is good cross-training for runners without the leg fractures you mentioned. You won't lose much, except for conditioning of the Calf muscles. When the doc gives the OK you can begin conditioning these gradually before resuming your normal workouts. You'll lose a little bounce and likely some aerobic fitness in the interim, but that's better than dealing with heel pain or Plantar Fasciitis the rest of your life.

You can see how your injury developed. The speed placed more reciprocal strain on the Achilles/Calcaneus. The frequency left little time for recovery. The distance exposed your heel to more impact. You have learned from this experience, and I thank you for sharing it with others.

A problem with biking could be over extension of your knees if the seat height is improperly adjusted.

If you have a question for your doctor, always ask.  You are paying the doctor for his time, knowledge and experience in order to resolve your problem.  If the doctor makes you feel uncomfortable for asking then find a better doctor.  Doctors are human and make mistakes.

Steve's right about shopping around if you are not comfortable with the diagnosis. Many docs have misdiagnosed Plantar Fasciitis as a heel fracture, if their training is oriented toward skeletal issues rather than musculoskeletal. The only reason I did not question the diagnosis is because you said the x-ray revealed a "fairly large stress fracture," which kind of does it for me. If you go elsewhere, see if you can get a copy of this x-ray for further interpretation rather than take more rads. It may be healing ahead of schedule, but it's important to know the cause of the pain. Here's why:

Plantar Fasciitis commonly results from the radical increases in mileage, frequency, or speed we were discussing earlier. More strain to the tissue = more likelyhood it will tense up. PF has also been micked by extremely tight calf muscles which pull on the Achilles, hence the heel. Ramping up your workouts could definitely cause this. If it happens with veteran runners, it would certainly affect the tender tissues of a new runner more. I know, because it happened to me when I did the same thing back in the day.

Another cause of outside heel pain is inflammation along the lateral tendons of the foot that pronate the foot. They can get cranked by forces that would otherwise twist the ankle on rough terrain, or due to sloppy high mileage running that comes with fatigue. Check the interactive tutorial below for the peroneal muscles (most of us have 3, some have 4, and some have only 2):

http://www.getbodysmart.com/ap/muscularsystem/footmuscles/fibularisbrevis/tutorial.html

Not sure about the pain thing, but I would not go by pain alone. Make sure you are really healed before you risk healing the wrong way. Chronic pain of any kind in the foot is the last thing a runner wants, and it can result later down the line from actions you take today.

Good luck!