This is perhaps a bit of a strange question, but it's bugging me no end. I know that it is acceptable to use both a singular and a plural noun after a plural possessive, e.g. "..this influences people's lives/this influences people's life" are both considered correct as far as I know... Now I have a sentence here which goes "... this influences students' mind" and it just doesn't sound right. Is that just me?
It's acceptable to use a singular noun after a plural possessive when the noun refers to a singular thing that is shared between multiple people (or animals, etc.):
We took our watch to the watchmaker. (There's one watch which we share ownership of)
We took our watches to the watchmaker. (Each of us has their own watch)
So grammatically, there's nothing wrong with any of the sentences you've brought up.
The reason why people's life is more acceptable than students' mind is that it's acceptable to consider life in some contexts to be "shared" between people (for example, you can write a thesis about the life of peasants in 15th century England, treating life as a total of experiences of all the peasants, rather than focus on individual lives of them). On the other hand, we usually treat each person as having their own mind.
That said, when it comes to abstract concepts such as life or mind that line is rather blurry. I personally wouldn't be very strongly opposed to students' mind.