Can an authenticated encryption scheme (like AES-GCM) detect if a wrong key is used for decryption? If not, what is the standard way to check whether the entered key is indeed correct. I presume KCVs can be used for this but does this somehow leak any information about the key?
This is an interesting question, and it depends on the situation where you might decrypt with the "wrong" key.
If two keys $k_1$ and $k_2$ are independently generated, and $c$ is an honestly generated ciphertext under $k_1$, then decrypting $c$ under $k_2$ will result in an error, except with negligible probability. If this weren't the case, it would lead to an attack against AEAD security (the attacker just submits a ciphertext under an independently chosen key). This analysis covers the case of "accidental" or "incidental" decryption under the wrong key.
However, this does not cover the case where $k_1, k_2, c$ are all generated adversarially. (Maybe an attacker shows you $c$ and $k_1$, and since $c$ decrypts successfully under $k_1$ you incorrectly conclude that someone with a different key could not have accepted $c$.) The usual definitions of AEAD don't prevent that. There are natural AEAD schemes (including AES-GCM) where it is possible to generate such $k_1, k_2, c$ such that $c$ decrypts without error under both $k_1$ and $k_2$. This property can indeed cause problems for some applications of AEAD, like password-authenticated key agreement and abuse reporting in encrypted messaging.
If it is hard to come up with any $k_1, k_2, c$ where $c$ decrypts without error under both $k_1$ and $k_2$, then we say that the scheme is key-committing. Sometimes the key-committing property requires providing some additional value (apart from the usual ciphertext and key) to help bind the key to the ciphertext. Key-committing encryption is studied here and here.