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3

Other than just deriving a ton of potential keys and checking whether they correspond to the address, no. There is no mathematical relation between keys in a HD keychain that allows you to determine whether they are in that keychain.


2

No. The computation from children from a parent is like a hash (and involves hashes). It doesn't matter how many children someone sees, they cannot compute the parent key. They cannot even tell which child keys come from the same parent. The only thing that's possible, and perhaps surprising, is that given a parent xpub, and a privkey derived from it, you ...


3

You need to ensure they use the same derivation path. If there are a lot of unused addresses generated you may need to increase the gap limit.


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