What can be learned from the ciphertext of LibSodium's crypto_box_detached()?
I'll answer the related questions in order:


No, because a ciphertext (generated from a key stream generated by a stream cipher)  should be indistinguishable from random data, and a MAC should be as well. 
No, because #1 depends on the secret, and the secret was derived using a Diffie-Hellman key agreement algorithm, using the given curve. To know information about the secret, it is required to obtain one of the private keys.
No, because that follows directly from #1 and #2.


Notes:


traffic analysis is not taken into account for this answer as it is outside the scope of the cryptographic properties;
the size of the ciphertext (equal to the plaintext size) + MAC size may leak some information to an attacker.




It seems that the detached variant was explicitly defined to separate the tag and the ciphertext, so it seems a bit strange to talk about ciphertext + MAC (only the order of ciphertext and MAC seems different from the non-detached variant of the same method).