Are all or any hash functions evenly distributed?

For example: if you had every combination of 256 bits and ran that through SHA256, would that produce approximately 1 of each combination, or would there be peaks?

Are distributions of the hashes calculable without going through the permutations in lower bit sizes?

Is evenness of the distribution of a hash function impacted by the length of the input vs the resultant hash length?

Which hashing functions are most evenly distributed? (e.g. as close to 1 of each equal-bit-length combination possible)

Edit: The linked "duplicate" doesn't come close to addressing all these questions.


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