As you found, the power absorbed by a fixed pitch prop (and thus the thrust too, more or less) is proportional to the rpm cubed.
This will be true for small changes e.g. a 1% increase in the rpm will increase the thrust by 3% (on the basis of the standard bit of calculus of small changes e.g. 1.01 ^ 3 is very close to 1.03).
For large charges, a lot of other stuff will come into it. A while ago I was reading something on propeller drag and saved
this article but while I was going to post the original URL I noticed that the original is now gone offline. Anyway this may give you some pointers showing that this stuff is significantly empirical.
There is a lot of stuff online e.g.
this.
I haven't got a clue how altitude affects efficiency but obviously it is an aerofoil "like any other" so thinner air will affect it. It has got to produce less thrust (and will be absorbing less power) in thinner air; that much is obvious if you consider the extreme case of no air at all. So in thinner air the AoA should be greater for the same effect, which brings us to the VP prop whose AoA is governed to (essentially) absorb a constant engine power output.
What are you trying to achieve?