Re: Martin Sax
For the octave key problem, I'm guessing you're missing a screw or a spring some place. My octave mechanism on my old (Conn) alto is very complex, but if you trace it well, you should be able to find what's missing.
For your other problem, you seem to use the term "key" to mean either "button" or "tone hole cap" without distinguishing. I will use the term "key" to mean what covers the tone hole, and "button" to mean that which the finger presses. I'm guessing you were trying to say that the g# key does not go down when you press any of the buttons on the right hand (which happen to be on the keys).
My first question. Does the key just above the F key go down with those three buttons? It should go down with each of them. If it doesn't, your problem is deeper than the g# key.
More likely, your horn is missing a bar that should extend from the key above the F button/key to depress the G# key any time that key closes.
Check to see if there used to be a bar there. If not, you may just have to lift your fingers off the G# buttons if you don't want g# to facilitate! That's probably why the extra G# key was developed in the first place, to make the key easier to maneuver with fingers One or Two, rather than the left-hand pinky.
I don't know enough about the Martin to know if that's endemic to the horn.