Thank you for your question.

Apparently birch tree leaves tend to turn yellow and fall off during hot, dry weather.  For healthy older trees, this is probably not a serious problem.  For trees that were planted in the last few years, it is more serious.   I'd water the tree deeply once or twice a week.  Here's a site to verify the information:
Some of the leaves in my river birch are turning yellow and dropping to the ground. Why? | Yard and Garden (iastate.edu)

As for the spots, they could be an indication of a birch leaf spot disease, a fungus such as Cryptocline betularum and Discula betulina.  This happens more often during a moist spring.  For this a fungicide should be applied in the spring, so there is really nothing to do at the moment.  In autumn, remove the fallen leaves.  
Birch Leaf Spot (uada.edu)

So, at the moment your tree does not seem at risk of dying!  From your description there are no indications of borers or other lethal indicators.

Hopefully it is old and healthy enough to survive these stresses. 
Sincerely,  

Thank You

Hi, I'm afraid I am not supposed to recommend certain brands, but since you know the name of two possible types of fungus 
(Cryptocline betularum and Discula betulina), any reputable nursery should be able to recommend an appropriate fungicide.  If you use one, you must follow the directions exactly.

I wish you the best! I love birch trees and have several!