I was going to answer, and then I realized that it is quite complicated and I don't really know what to say, despite being a native.
At least I can say this: "nummer" is used for phone numbers, your place in a queue or you house number, and generally for things that might as well be a name and doesn't necessarily count something.
Treat "hvor mange" as a single unit meaning "how many", just as "hvorfor" means "why" and "hvordan" means "how"
In "hvor mange" is just not written in one word like the others are.
I believe hvordan is more of a "How did you do that?" and hvor is "how close is it?" (one is about an action, the other is asking for a description).
"tal" doesn't have a plural form? or it's not required by the "hvor mange" question structure?