I have a chimney and mine too during heavy sideways rain leaks down the inside of the flue and then it lands on the smoke shelf then drips down into the actual ashes and makes that wet ash smell.
I too have an oversized chimney cap and it assisted me tremendous, but not 100%. I have just learned to live with this.
The anatomy of the brick flue fireplace shows the smoke shelf being errected in a manner that directs water down to the back of the fireplace and not in the actual box, if this is not done right during construction, it will leak.
Now, the leaking in other spots, I would spec that its your metal flashing around the actual exterior chimney making the water tight bond between the home and the brick. I would suggest on top of all that you have done is to actual hand caulk with "butyl rubber" caulk around all those ninety degree flashing things. This is a pain for you and I understand it, but these things happen between the brick and the home.