1 - If X is $\sigma_1$-subgaussian and Y is $\sigma_2$-subgaussian (not necessarily independent), then is it true that X+Y is $(\sigma_1 + \sigma_2)$-subgaussian?
I got $\sqrt{2 \sigma_1^2 + 2 \sigma_2^2}$-subgaussian by applying Cauchy-Schwarz inequality on moment generating function, but I couldn't figure out something anymore.
2 - For any random variable X and Y, can we say $Y-E[Y|X]$ is independent of X?
I heard that $E[Y|X]$ is a kind of 'projection' of Y onto X, so can we say $Y-E[Y|X]$ is independent of X because $Y-E[Y|X]$ is 'orthogonal' to X?
The first part is correct but can be tightened using Holder's inequality instead of Cauchy Schwarz:
$$\mathbb{E}e^{t(X+Y)} \leq (\mathbb{E}e^{tXp})^{1/p}(\mathbb{E}e^{tY(1-p)})^{1/(1-p)}$$
Then applying the known subgaussianity and optimizing $p$ to minimize this upper bound we get $p = \sigma_2/\sigma_1 +1$ giving that $X+Y$ is $\sigma_1 + \sigma_2$ sub-gaussian.
$E(Y- E[Y|X])g(X) =0$ for all measurable $g$ by definition, so they are uncorrelated but not necessarily independent.