Applying to work at an acquaintence's school
How do you go about telling such an acquaintance that I am planning to or have already applied? Is it acceptable for me to simply drop them an email? Is there anything specific I should tell them?


Yes, you should just send them a short email (don't make it too long, or write it in a way so that the recipient is pressured to answer it). I would email them after all of your application materials are in, and tell them that you have applied to their school. If it is among your top choices, I would mention that in the email too.


  Most schools list a designated contact person within the department.
  Would it then be inappropriate for me to ask my acquaintance questions
  about the school or position?


The contact person is most likely the secretary of the department, or the head of the hiring committee. However, every member of the department has some say in the hiring process, and it is generally better to establish contact with your acquaintance. However, I would hold off on asking questions about the school or department until you receive an interview, unless the answer to the question that you are asking decides whether you will apply to the school or not. If there is no chance that you will be hired, then you don't want to annoy your contact and waste both of your time.


  One of these acquaintances works at a school that is far from the
  region I would like to end up living someday, and I believe that they
  are aware of this. How will this impact my chances at that school?
  Would it still be beneficial to make them aware of my application?


This seems like a tricky question. If you are a superstar, they might still try to interview you and convince you that their school is fantastic. However, if they are deciding among several candidates of similar calibers, and if one of the candidates rank the school as their first, that could tip the balance in their favor. Nonetheless, it wouldn't hurt to alert them of your application. However, do not lie and say things like their school is your top choice. In a small world like academia, you do not want to burn any bridges if you can help it.

Finally, I'd like to add one thing out of paranoia. Do not apply to positions that you would not accept. This is especially important at tenure-track level, where there is a lot of work that people have to put in before an offer can be made. At the tenure-track level, once you receive at least one offer, it is expected that you accept one of them, otherwise you may not be taken as seriously when you re-enter the tenure-track market in a later year. These kinds of things get talked about, and as the job market is a sensitive issue for most mathematicians, people tend to have very long memories about these sorts of things.