Hash 22 byte secret into 512 bit hash: Full entropy? Does it make sense to hash a 22 byte secret (which may not have full entropy since it is chosen by the user) into a 512 bit hash (using Keccak-512)?

More precisely, will the 512 bit hash have full entropy even though the input is only 22 bytes?
Yes, the 512-bit hash (using Keccak-512 or the like) of 22 bytes (176 bits) entered by a user (or from whatever source) has the full entropy of the input (that is: at most, 176 bits of entropy; much, much less for a user-specified 22-character string) for any practical purpose; and not as much entropy as there are bits in the hash's output. Argument that there is no reduction of the original entropy: odds that among the $2^{176}$ 176-bit strings, any two hash to the same value, are only $\approx 2^{-161}$, making the operation theoretically reversible (including for every input) with overwhelming odds. Arguments that entropy is not increased to the hash's width: the number of possible outcomes is no more than $2^{176}$ 512-bit values, which are fully determined by the specification of the 512-bit hash; and no deterministic process can increase entropy.

In fact the reduction in output entropy compared to input entropy remains mostly negligible for input of 63 bytes (or less); or more generally for input of at most $n-1$ bytes for an $n$-byte hash. Argument: even though the transformation is no longer reversible for every input, it still is expected to be reversible for a large majority of inputs.

This justify Owlstead's suggestion of using a narrower hash, such as SHA-224 or SHA-256, since the output will be smaller but have almost the same entropy.

However it makes only limited sense to hash a user-specified string: since the input is chosen by a user, it is vulnerable to guessing, and replacing that input by its hash won't solve that issue. For this purpose it should be used a key stretching mechanism, such a scrypt (or the more common but much weaker PKBDF2). Key stretching does not increase entropy, but greatly increases the difficulty of brute-force search (about: by the ratio of effort spent for stretching, to the effort necessary to decide is the stretched key is correct; that can be $2^{30}$ in some realistic situations, like a password-protected document).