Re "Do I have to treat these things as a fact or there can be some logic behind it?" See the math behind the Heaviside step function and the Dirac Delta Distribution.

The frequency does not change in time in a linear system. It might in a nonlinear system.

should "flux in the x direction along the area S" read "flux in the x direction through the area S" ?

a little red ball with a minus sign on it ??

Fields that fall off as 1/r have an infinite discontinuity at the origin (r=0). How is that handled? E.G. a point charge at the origin.

@naturallyInconsistent Re. "It is just a mathematical trick for finding where the next wavefront will be". Not a trick but rather a model of wave propagation. It is a good model that is related to a Green function approach.

@Michael Seifert but at the two ends of the segment the string is stretched?

Intermediate Value Theorem can be found where?

so then the wall would experience no 'net force?'

What would the medium be in a vacuum?