Hashing functions in general only reduce a long variable-length input string to a short fixed-length output, while preserving as much information as possible within these constraints, so that input strings that differ in some manner that is considered important for the intended application are hashed into distinct values.
So if one wants a non-secure hash that preserves some kind of distance defined on the input strings, that is possible in many cases.
https://en.wikipedia.org/wiki/Perceptual_hashing
Non-secure hashes are not useful for detecting data modifications, but they are useful for searching through large amounts of data, especially when the search is not only for identity, when secure hashes can also be used, but also for similarity, when secure hashes are useless.
I don't think that's possible. If you can hash a test image, and you can evaluate its "distance" from other images, an attacker can use that distance as a metric for a hill-climbing or simulated annealing algorithm.