We read $$\sup_{x\in X}(\sup_{y\in Y} f(x,y))$$ left to right (there's really no other choice), but Question 1: doesn't this contradict the following conventions?
- Stuff inside parentheses is evaluated first
- Composed functions (such as $\sup:X\to\mathbb{R}$ and $\sup:Y\to\mathbb{R}$) are read right to left
I guess one could argue 2. doesn't even apply, since they aren't composed functions at all (Question 2: what are they, actually?), but the entire thing still feels weird to me, especially since this entire thing seems to be understood implicitly and not defined explicitly.
I have no idea what you're talking about with convention (2); I'd "read" the expression $f\circ g$ as "f composed with g", from left to right.
Regarding the supposed contradiction with reading left to right and convention (1): No, there's no contradiction there; I can't imagine what you think the problem is. Consider a simpler example, the expression $$\sin(t^2).$$This is "read" from left to right:
So what? The fact that we pronounce the word "sine" before we pronounce the word "squared" has nothing to do with how the expression is to be evaluated - we need to evaluate $t^2$ first, so we know what to take the sine of. Math is the same on the planet htraE, although they read the expression as