AES GCM : is it acceptable to return the wrong plaintext if the tag is incorrect?
There is an article* that answers the question in the negative for GCM and CCM. The article introduces the first formalization of the Releasing Unverified Plaintext (RUP) setting. The related security notion is the Ind-RUP.
The security question is can an adversary forge messages with unverified messages? In this game, confidentiality is not relevant, since in the game the adversary is accessing the plaintext of the chosen messages. AES-GCM fails to satisfies this.

How to Securely Release Unverified Plaintext in Authenticated Encryption by Elena Andreeva and Andrey Bogdanov and Atul Luykx and Bart Mennink and Nicky Mouha and Kan Yasuda


AE schemes such as GCM [34] and CCM [49] reduce to CTR mode in the RUP setting. This is because the adversary does not need to forge a ciphertext in order to obtain information about the corresponding (unverified) plaintext.


Your questions:

It is very dangerous. AES-GCM internally uses CTR mode and bit flipping is very easy in the CTR mode. An attacker can easily change the ciphertext bit on their advantage. This is catastrophic if the structure of the message is known.
One must never ignore the tag invalid!
It is not about the leak of the key, CTR mode is IND-CPA secure. It is about changing the messages. This is why CTR mode can't have IND-CCA. GCM provides integrity and authentication and authenticated encryption &gt; IND-CCAs

For the second part you may benefit from is Q/A which provides a solution for the limited chunks for the GCM;

How to securely encrypt/decrypt data with a maximum chunk size?




Also for general guide

What are the rules for using AES-GCM correctly?, and
Deep into bounds Does GCM (or GHASH) only provide 64-bit security against forgeries?


*Thanks to Squeamish Ossifrage for pointing the article.