In English it seems that we can only have partitives of definite nouns; i.e. you can have Some of the dogs but you can't have *Some of dogs.
This explains why you can only use quantifiers that can denote a partitive sense for this construction, like some and many and few, and not articles or possessives or other adjectives.
I'm not so sure. "All" and "none" can be stacked, as can the non-quantifier "each." "Each dog" and "Every dog" is synonymous, but with "each" you can make constructions like "Each of my dogs."
Secondly, you can make constructions like "Part of a dog" so I'm not sure if these things are restricted to definites.
I would prefer this analysis especially because you can double-stack, like Adam's many dogs. You can't double-stack some, however, (*Adam's some dogs), but that's probably because some already denotes a partitive sense. Without the of, though, some can just mean "a few" or somesuch.
"Many" and "Few" can be used as adjectives in this sense and yet they also denote partitive relationships. I think the "many" in "Adam's many dogs" and "Many of Adam's dogs" should be counted as two totally separate lexical entries rather than a single entry that is somehow capable of doing both.