While calculus of variation from which Lagranges equations of motion arises states integral to be an extremum, it seems almost always to yield the minimum. Is it simply way things work or is there a mathematical reason to it?
Also, is it possible for a solution to Euler's equation for a given system to yield expressions for both minimum and maximum? If not why is that?
For starters, the stationary path for a functional is not at a minimum if there are conjugate points along the path. This is the generic situation and happens already for the harmonic oscillator, cf. e.g. my Phys.SE answer here.
By changing the sign of the functional, a minimum can be turned into a maximum, and vice-versa.
In physics applications, there are rarely both a minimum and a maximum path.