Any 3 points define a plane and therefore ALWAYS lie on the same plane. The same goes for two points and a straight line.
Does this mean that a hyperplane in four dimensional space is defined by any four points? Can the concept be applied to higher dimensions and still make sense or be useful?
Yes, if they are not all in the same 2-dimensional plane. (The same way that 2 point determine a line if they're not equal, and 3 points determine a 2-dimensional plane if they're not all in a line.)
In general, in a space of dimension at least $n$: if you pick $n+1$ points in the space, they will determine a unique "plane" of dimension $n$, unless they're all in the same "plane" of dimension $n-1$.
Yes. These things are typically taught in a course in Linear algebra. In linear algebraic terms, an $n$-dimensional "plane", as I was referring to it, is called an $n$-dimensional affine subspace.
Linear algebra is arguably one of the most fundamental pieces of math in use today, so it should certainly be considered useful.