Is it ever OK ...
To contact HR to see if your application was received? Yes, by email or by phone, but keep a phone call brief.
To contact the department or search committee chair to see how things are proceeding? A short email is fine, a phone call never. Don’t be insulted if the chair doesn’t reply, and definitely don’t take it personally. It’s bad manners, but chairs are incredibly busy.
To mention that you met someone in the department five years ago? Yes, but I’m not sure what it will do for your application. Advanced scholars in particular meet many people in a five-year period, and a quick introduction would be unlikely to be remembered. If the person does not remember you, you should not take personally.
To mention that your parents/in-laws/best friends live in the area? Yes, and the first two items might actually have some bearing if a department has had a problem with people leaving positions after a year or two. This would be particularly true for a smaller institution’s openings.
To ask a trustee or major donor to contact the search committee on your behalf? Never. That's always a bad idea. Nothing good can come of it.
To update your application post-submission? Generally, no. If you land a book contract or a first-tier publication, just send the search chair or department chair an email. Otherwise it looks overly fussy.
To correct an error on your application post-submission? If you just submitted it and it has not likely been reviewed, this is fine, particularly if you realize you sent the wrong application or you've made gaffes such as forgetting to change the name of the institution from a previous version. Just ask the contact person to swap the two files. If you realize you've made an error very far into the process, you are stuck with it once it's been seen. Pulling a file at that point just draws attention to the error.
To check on the timing of a decision because you have received an offer elsewhere? It’s your call: Sometimes searches are sped up in order to be more competitive. So definitely email the department chair about this, especially if you are past a phone interview. In some cases, however, nothing can be done early in the process, and if the search committee believes that you are likely to be out of consideration, they won’t continue to include you in their deliberations.
To ask why you weren’t hired? No. I’m sure you’d appreciate the feedback, but search committees cannot and should not discuss their deliberations outside of those meetings.
What other questions and / or answers would you add?