This question as formulated may be opinion-based. Still, I'll take a whack at it.
Should professors have deadlines for returning graded assignments (beyond just the end of the semester)?
Should is a difficult word. Personally, I've long felt that both instructors and students should try to do things right away when possible. There is never a perfect time to do/grade an assignment, and attempting to do it right away will prevent oneself from becoming inundated by deadlines. Both instructors and students seem to have an irrational desire to postpone doing unavoidable work until "the last minute" (or later), which I've never understood.
As a practical matter, I think a policy requiring this would be very hard to enforce. At a major research university (so-called "R1s"), professors are researchers first and teachers second. They have many deadlines: grant proposal deadlines, proposals and reports related to their grants with hard deadlines, presentations that must be prepared for, lectures that must be prepared for, meetings with collaborators at which one must show results, etc. Trying to set a "strict" policy (e.g., return assignments within one week) would rightfully lead to some pushback. Setting a "soft" policy (e.g., return assignments within a week unless you're busy doing something else) would be meaningless; everyone could up with some excuse for being late, and no one wants to tell their colleagues how to manage their time.
If so, what would reasonable deadlines be for different assignments?
As a rule of thumb, I would say that papers and exams should be returned at least a few days before the next one is submitted. Homework should be returned within a few weeks. But again, I am only proposing this as a guideline.
I realize that waiting to grade assignments does give the professor a chance to show mercy to students who sometimes turn in their assignments late
It may make sense for the professor's grading timetable to be intertwined with the late policy. I've suggested this here. But in other cases, the schedule / late policy may have been designed with other considerations in mind, and so these two things are completely unrelated. For example, in a foreign language course, it is essential that students learn the current grammar/vocabulary before moving on, and so due dates should be quite strict regardless of how far behind the professor's grading is.
I also know that some larger universities employ teacher assistants to manage this part of the job, so how does this effect the issue?
Yes, the above reasoning doesn't apply so much to teaching assistants, and perhaps also not to faculty whose primary job is instruction (e.g., at teaching colleges or community colleges). Naively, it seems like stricter policies would make more sense for these institutions (though it's quite possible I'm wrong about this; perhaps someone with more experience at such institutions will correct me).