Is there any way to know that the private key is more or less than n/2 while not knowing the private key itself?
Again, no.
Is there a way to find out the public key of an address that never sent Bitcoin but only received it?
Depends on the address. If the address is a hash of the public key, such as in P2PKH or P2WPKH, then no. If the address is not a hash of the public key, such as in P2PK or P2TR, then yes.
I think if a Taproot spending path is used, then only that path will be revealed in the BIP141 (the witness program BIP)-style witness, not the public key since it would be excluded in this case, right?