High Heart Rate when Exercising

I am an active 30 year old female.  My resting heart rate is generally between 70-90, which is higher than a lot of people I know.  Additionally, I have been diagnosed with hypothyroidism (currently treated), and more recently, severe Vitamin D deficiency and low estrogen levels.  

One of my goals has always been to be able to jog a 5K without stopping to walk.  I have never been able to run long periods of time without losing the ability to breathe appropriately.  I have exercise induced asthma as well, but I don't feel like the asthma is what stops me from running.  My heart rate gets over 200BPM within 3-5 minutes of jogging, and I am unable to maintain that level for long without feeling shortness of breath.  My doctor has told me not to worry as long as recovery is quick, and I am usually back in the 150-160 range within a minute or two.  However, I feel that this issue is keeping me from reaching my running goals.  Is it bad for me to push through at this 200+ level??  I don't want to put myself at risk for a heart attack, but really would like to keep pushing toward my fitness goals.  

[ed34]

well some cycling athletes reach over 200bpm but it depends on why your heart needs to climb that high. With running I would have thought this would be too high. However, saying that, when in cardiac rehab after my triple bypass my heart was like yours. In just 20-30 seconds it would jump to 180+ but with no discomforts. I felt good. When I slowed down, it would drop to 90 in around a minute. The cardiologist said this was great but cardiac rehab staff used to panic.
With such a high heart rate, Im surprised you are getting short of breath. Are you on the right diet to give you the energy required? and hydrated?
Wouldn't vitamin D deficiency cause a problem with the muscles? causing a problem with your fitness?  I think vitamin D is mostly made in the skin using sunlight? and this affects muscles/bones.
I don't know how long you have been exercising this way, obviously if you have just started recently I think maybe your goals are perhaps a little too high. Start with smaller distances and work your way up gradually. When I started running (many moons ago), I began with just half a mile. Although this was easy it gave me a gentle starting point to build from. After a couple of weeks at a time I would add a quarter of a mile. I reached 9 miles every day in the end but that was enough for me, especially before going to work.
I made sure I had light healthy breakfast an hour before starting and water.