Welcome Patrick,

> Is there a book anyone would recommend as a synopsis of Buddha's
> teachings on the core issues of his philosophy based on the Pali
> texts?

I would recommend "The Buddhist Path to Awakening" by Rupert Gethin.

http://www.bris.ac.uk/thrs/staff/rg.html
http://www.amazon.com/Buddhist-Path-Awakening-M-Gethin/dp/1851682856
http://www.wisdom-books.com/ProductDetail.asp?CatNumber=4899

> Does anyone know if there are any Buddhist organizations or sects
> today which rely exclusively on the Pali texts for their beliefs and
> practice?

There are. You can find some organizations in the Sadhu! directory
http://dhamma.ru/sadhu/

Good luck, Dmytro