I have had a sharp pain in my heel. If you looked at the back of a cycling shoe and picked a spot half way between the sole and the top that is where the pain is. I feel it for the first 3 -4 minutes of my runs and then it mostly goes away. It is also present first thing in the mornings. Stretching my calf muscle does not seem to solve the issue. When I look at the anatomy of the foot, the pain seem to be at the point where the Achilles Tendon attaches to the heal bone (Calcaneus). There is a Bursa sack in that general area. Has anyone heard of Bursitis of the heal?
Yes, I’ve had it of and on for a few years. Doc wanted to inject it, I really didn’t want to go there. I was running in orthotics at the time and I think the heal cup in the orthos aggravated it. Quit the orthos, pain seemed to go away. Pain free for a couple seasons now, even though last year I was back to using the inserts.
it could be retrocalcaneal bursitis, it could also be insertional Achilles tendonitis… which also commonly occur together, so you might be so lucky as to have both 
I’ve had it for about a decade now, last 3 years it’s been incapacitating. The most effective treatment has been flat eccentric heel drops,
http://www.runningwritings.com/2011/09/injury-series-flat-eccentric-heel-drops.html
After six months of these the pain is manageable after running.
For the morning stiffness, try a Strassburg Sock, or night splints. I have a night splint which has been helpful, can get up and walk instead of limping…
Background reading here,
http://www.takethemagicstep.com/coaching/athletes/health-management/managing-achilles-tendon-injuries/
Thanks. That second link especially was helpful. I did ramp up my bike riding this fall which had the benefit of improving my calf muscles but they always felt tight even after rolling/stretching. Since the calf muscle is attached to the Achilles Tendon, it may be Insertional Achilles Tendinitis. I will have to try the exercises at that link and see if they help. I tried the sleep socks but for me they prevented sleeping. I started pointing my toes toward my knee for a few moments as soon as I wake up and that seem to make the first few steps more manageable.
Inject with what? Cortisone?
Yes.