Are there parts of a SHA-512 hash that occur more often then others? Suppose I have a 'begin' password that I can remember.

I calculate it's SHA-512 hash, transform it to base-64 and check if it has a least 1 lowercase, 1 number, 1 capital, and 1 sign.

If this is not the case I repeat this procedure (but now use the previous hash as input) until I have a hash that fills these requirements.

I will now use this final hash as the 'final' real password.

Will every possible combination of bits be (approximately) equally likely?
(If I ignore the first lowercase, number, capital and sign in the password and only look at the other chars)

In short: Will every possible 'final' password be equally likely?
In short: Will every possible 'final' password be equally likely?


Yes, we expect that they equally likely. Cryptographic hash functions expected to have a well-distributed output. If you find any bias, you can convert this into an attack. Even there is a tiny bias then that can be used to attack for the hash function.

You seem like you are going to use the SHA-512 hash function as a password generator. Your scheme's weak point will be your master password. One can still write a password cracker for your construction. And Cryptographic hash functions are not designed to be slow, they are designed to be fast. That is why we use iteration on Key Derivation Functions. A better solution using PBKDF2 or Argon2, and use their output. You can use the iteration number as the next try if not occur. So first look at, 100000 then 100001, etc.