Can they do this to me?

What is the name of your state (only U.S. law)? Oklahoma
I was in the hiring process (firefighter) with a city and even got a conditional job offer pending the outcome of a health physical, which I passed, when I was charged with a felony. They sent me a letter saying that I was no longer being hired due to the pending charge. The charge was ultimately dismissed...According to this cities policy I am now too old for this position. Do I have a case?

Did you have a birthday? Is there a legitimate health reason that the position has an age cap? What is the age cap?

The age limit is 44 yrs. I did have a birthday turning 45 during this whole process...

Also, the state fire pension system has this policy that the maximum age to be 44 yrs of age prior to being hired. I guess it has something to do with being able to retire before the age of 65...No health reason.

I'm not sure that's a legal requirement. Talk to the EEOC.

What part of the physical exam led to the felony?

The exam had nothing to do with the charge. I was charged on an unrelated event and it was then dismissed...