Whether using a fixed IV is acceptable depends on the mode of operation you're using. Some modes only require that the IV be unique under a given key, such as CTR mode, while others, such as CBC, require an unpredictable IV. Attacks on CBC are known with predictable IVs.
The adaptive chosen plaintext attack mentioned exposes the plaintext, but not the keys. In general, a secure block cipher itself should not be vulnerable to exposing the keys even if some correspondence between the plaintext and ciphertext is known, which is usually what happens with IV misuse. I'm not personally aware of any common AEAD modes which require an unpredictable IV, but in general IV misuse with AEAD modes can compromise both confidentiality (expose the plaintext) and integrity (permit tampering).
The easiest way to avoid reusing an IV in this way is to take the random value you would normally use for the key and run it through a key derivation function (KDF), like HKDF, to derive both a secret key and an IV. If you're already using a password-based KDF to generate the key, you can also generally generate the IV using the same KDF. Since this is so easy to do, there's little reason to reuse an IV in a practical environment.
