After sending a faculty application online, should the applicant send an email to check if all documents were received and whether the application is complete or not? Some suggest to send an email, while others suggest not to, since sending unnecessary email to the committee could raise a flag.
Assuming that you submitted your application to the human resources office of the university, then it would be appropriate to check with them to make sure that the application was received and was considered to be complete. I would not send a query to faculty members of the search committee (if you could even figure out which faculty were on the search committee.)
I think it's okay. Just make sure you are emailing someone who knows where to get the information you seek. Otherwise, it may "get buried" and disregarded.
Email is an accepted and efficient method of communication and should not offend somebody's sense of etiquette. If I were the one receiving your email, I wouldn't be offended in any way unless you were bugging me frequently.
Just a couple months ago, I received an email request from a friend to be a reference for him. I thought nothing of it.
Try to find out the staff person who is responsible for helping the search committee. Usually this is the person who you addressed the initial file to (our senior admin here is used to getting e-mails addressed to Prof. Lastname, it tickles her sense of humor).
The staffer usually has the best sense of where things are in process. All of the files are at his or her fingertips, whereas a member of the search committee may not be able to recall all of the first-round cuts and the search committee chair may be too overworked or stressed to respond. If your file was incomplete, the search comm members may not have even seen it.
Annoying the staffer has little influence on your file, so that person is the safest person to bother with questions like this.(FN1)
FN1: I won't say it is entirely risk free. If you are truly obnoxious to our staff, then the faculty may view this is as a sign of your future collegiality. So act within reason and professional behavior.