Does Grover's algorithm (or any other applicable quantum algorithm) only apply to weakening the hash function itself or can it also search only plausible inputs (a-z, A-Z, 0-9 <=14 characters) to reduce the search space?
Grover's is a general search method - it doesn't know or care how the function is given interprets the input. If you give it a function that interprets the inputs as "(a-z, A-Z, 0-9 <=14 characters)", it'll work with it, with the expected complexity.
On the other hand, the relevant question may be "is using a Quantum Computer make economical sense, as opposed to a GPU farm or some ASICs?" I suggest for the next several decades (that is, until Quantum Computer technology has gone through a number of generations), classical methods will be faster in this instance.
Some of the reasons for this:
The cost of a Quantum Computing operation can be expected to be much larger than the cost of the corresponding classical one. This is a constant factor when comparing the two - however, if this constant factor is perhaps a billion or a trillion, it really shouldn't be ignored.
Parallelization - classical computer searches (like this one) are perfectly parallizable (which a GPU farm or set of ASICs would take full advantage of). In contrast, Grover's algorithm (or any similar Quantum Oracle search) is not - you do get that the $n^2$ speed up if you can perform $O(n)$ successive operations. However, if you can't wait that long and need to implement multiple Quantum Computers to do the search, well, you don't get that $n^2$ speed up between them. For example, to do the search 100 times faster, you need 10,000 times as many Quantum Computers.
Reuse of the key search. With classical computation, we can generate a Rainbow table - generating that table takes about as long as the full search; however once it is done, doing repeated searches on various hashed passwords is cheap. In contrast, running Grover's algorithm will do a search on a single password - if we want to check on another hash, we have to pay for the full search all over again. You could build a Rainbow table on a Quantum Computer, of course; however I don't believe you'd get any Quantum speed up, and I haven't heard of an alternative where you would get a Quantum speed up.
Now, over time, we can reasonably expect the cost of Quantum Computers to go down (and go down faster than the comparative cost of classical computing), and we can expect Quantum Computers to get faster over time (hence requiring less parallelization); I just don't expect either trend to happen over night...
To be clear, I understand that any bare hashing function (even with unnecessarily long outputs like SHA-512) is a terrible way to store user passwords and Argon2 or similar should be used, but I thought that memory-hardness and other aspects might make the question too broad or an answer less broadly applicable.
With our current understanding of the trade-offs inherent within a Quantum Computer (which might be considerably off-base), a memory hard function would be quite painful for a Quantum Computer to deal with. Evaluating such a hash function would appear to require a large Quantum memory, and (as far as we know) that looks to be dreadfully expensive [1].
Grover's algorithm would be applicable (again, it doesn't care what the hash function is); however it would appear to be quite expensive.
So, assuming that our guesses are correct, a memory hard hash function (Argon2) would be even more resistant to Quantum Computing than simple hash functions.
[1]: I believe at least part of the reason is if you have a large Quantum Memory, and access it for an entangled address, you end up performing operations on every single memory cell (or at least, every single cell that might be addressed by the entangled address). In contrast, with a classical memory, you only need to access the cell which is being addressed. If we're talking about a memory with 1,000,000 addressable locations, this means that the Quantum Memory might need to perform 1,000,000x as many operations compared to the classical case.
