Normally you change medium at a fixed place and continuously in time. If you could change the index of refraction of an extended area, but at the same time, then the wavelength would stay fixed and the frequency would change. A cute thought problem, but I'm not certain if it has ever been done...
Hmm, I don't know. Have you gotten it working from the same direction? You are definitely using two halves of a split beam, not two different beams? How far do the two beams travel from split to recombination? Can you see light scattered on the screen?
How would you measure that? Usually they are coming from the same direction so they can simultaneously be projected onto a single screen. A standing optical wave, (which I believe has been made), is basically interference of counterpropagating waves. Actually, there is a case of...