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You're absolutely right, it must be $\Delta f = 1/T$, since this is the base frequency of the fourier transform of periodic functions with period $T$. Otherwise, it's clear for $m=3$, for example, that the exponential would become $g(t)=\exp\left(j \frac{2\pi}{2T}\cdot 3 \cdot t\right)$ So it does not obey periodic boundary conditions at the right ...
You got some definitions wrong. It's correct that orthogonality means that $E[XY]=0$. Uncorrelated means that $X-\mu_X$ and $Y-\mu_Y$ are orthogonal, i.e., $E[(X-\mu_X)(Y-\mu_Y)]=0$. If you work that out you should arrive at the equivalent condition $E[XY]=\mu_X\mu_Y$ for uncorrelatedness (not for independence!). Consequently, if at least one of the two RVs \$...