Are non-US citizens elegible to sign up for TSA-Pre?
My understanding is that it's a more complex process, but once you're processed, it's much faster to get through airport screenings in future. Is it available to non US citizens as well, and if so, once approved, how long are you approved for?
Yes and No.
There are a few different ways to get TSA Pre - each with it's own restrictions.
You can get access via your airline, particularly if you're a frequent flyer. Historically you had to specifically request to gain access via this means, however many airline now will automatically submit you for access. Check with your airline to see if there's a specific process, although this is likely only for US airlines. You do NOT need to be a US Citizen or Permanent Resident to get access this way (historically you technically did need to be, but it wasn't well enforced, and this restriction has recently been removed).
You can get access via Global Entry, SENTRI, or NEXUS. These are programs generally designed for frequent travellers to make crossing specific borders (specifically the US/Canadian border) easier. Some non-US Citizens are able to join these programs, especially Canadian citizens. You do NOT need to be a US Citizen or Permanent Resident to get access this way, however access is only available to citizens from a very small list of countries.
You can get access via the TSA Pre Application Program, where you basically pay a fee to be screened and then given access to Pre. Currently you DO need to be a US Citizen or Permanent Resident to get access via this means.
If you are over 75 years old, or under 12 years old you will be given access. You do NOT need to be a US Citizen or Permanent Resident to get access this way.
As an Australian with high frequently flyer status with United Airlines I get Pre access the majority of times I fly (and have pretty much since the Pre program was introduced).
They might also randomly add it when in a group, I think. I was recently travelling in a group of three (one 70, one 77, and one 35) from international to various domestic US locations, and the 70-year-old had a TSA-Pre logo on all domestic passes except the final transfer before the international departure. No application necessary; no idea what triggered it. None of the travellers were US residents.
Very small clarification: while you can get NEXUS as a Canadian permanent resident that does not make you eligible for TSA Pre until you get citizenship. I know this first hand, I myself am in this middle. I will go on a limb and say this is because of the problems of verifying Canadian PR status outside of Canada.
Since April 2015 you must have Known Traveler Number to enroll to TSA Pre, which means you have to be a citizen or a permanent resident - http://blog.tsa.gov/2014/06/tsa-travel-tips-frequent-flyer-profile.html