1. "Which child is missing, did you say?" "The child whose mother is a teacher."
Alternative (though slightly odd) meaning:
1a. "Which child is missing, did you say?" "The child that the mother teaches."
With #2, it seems you have to lose "some", if you use "whose":
2. "Which book did you throw away?" "The book whose pages were missing."
To keep "some", you could say:
2a "Which book did you throw away?" "The book of which some pages were missing."
(But it sounds very stiff and awkward.)
3. "Which boy broke the window?" "The boy whose sister I visited."
(Which sounds a very dubious arrangement.)
I'd also agree that the original sentences (though awkward, and probably non-standard) might turn up in conversation, since we quite often find we have to string qualifying clauses after a noun, if we open our mouths too soon. (When I say "we", I mean "I".)
All the best,