Is it possible (and recommended) to also store SSH keys in Ansible Vault?
Yes. You can encrypt string values or entire files using vault.
On a more pedantic note, you don’t store things in Ansible Vault. Vault is just a command line tool used to encrypt or decrypt values. It is similar to encoding or decoding something with base64 (obviously, vault is more secure)
Is it safe to push a Vault-encrypted file to Git?
It uses AES256 for encryption. It is very secure; I think storing secrets in git was probably a big use case when they created it