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.
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 ...
In BIP32 key derivation, having the parent extended public (xpub) key suffices for computing child public keys.
That parent xpub can be derived in a number of ways, but all eventually boil down to a seed. However, that seed is only used indirectly to construct the xpubs. Once you have the xpub, you can compute the child keys.
No, it's not getting significantly easier.
The key space is literally 67 magnitudes larger than the count of all addresses used so far. Even if address use increases by magnitudes it would make effectively no difference.
Bitcoin has no mechanism to prevent users from rediscovering keys except the vastness of the key space itself, so just because keys are ...