I am used to ... that people are trustworthy
First, changes to English grammar in the 19th Century made "the fact that" a ubiquitous, if somewhat ugly, phrase that we need to use in certain grammatical situations. See Ngram. Don't worry about using it; it doesn't grate on native English speakers (although I suppose it might have in the early 19th century).
Fortunately, this isn't one of these situations. Your third suggestion,
I am used to people being trustworthy,
means nearly the same thing and is less ugly. The first and fourth are also grammatical, but the meaning isn't as close to your original.