How to calculate test error

I’m reading about test/generalization error in Hastie et al.’s Elements of Statistical Learning (2nd ed). In section 7.4, it is written that given a training set $mathcal{T} = {(x_1, y_1), (x_2, y_2), ldots, (x_N, y_N)}$ the expected generalization error of a model $hat{f}$ is $$Err = E_{mathcal{T}}[E_{X^0, Y^0}[L(Y^0, hat{f}(X^0))|mathcal{T}]],$$

where the point $(X^0, Y^0)$ is a new test data point, drawn from $F,$ the joint distribution of the data.

Suppose my model is a linear regression (OLS) model, that is, $hat{f}(X) = Xhat{beta} = X(X^TX)^{-1}X^TY$, assuming that $X$ has full column rank. My question is, what does it mean to (1) take the expected value over $X^0, Y^0$, and (2) take the expected value over the training set $mathcal{T}$?

For example, suppose $Y = Xbeta + epsilon$, where $E[epsilon]=0, Var(epsilon) = sigma^2I.$

(1) Consider evaluating $E_{X^0, Y^0}[X_0hat{beta}|mathcal{T}]$, is the following correct?

begin{align*}
E_{X^0, Y^0}[X^0hat{beta}|mathcal{T}] &= E_{X^0, Y^0}[X^0(X^TX)^{-1}X^TY|mathcal{T}]\
&= E_{X^0, Y^0}[X^0|mathcal{T}](X^TX)^{-1}X^TY\
&= E_{X^0, Y^0}[X^0](X^TX)^{-1}X^TY
end{align*}

The last equality holds if $X^0$ is independent of the training set $mathcal{T}$.

(2) Consider evaluating $E_{mathcal{T}}[X^0hat{beta}|X^0]$, is the following correct?
begin{align*}
E_{mathcal{T}}[X^0hat{beta}|X^0] &= X^0 E_{mathcal{T}}[(X^TX)^{-1}X^TY|X^0]\
&= X^0 (X^TX)^{-1}X^TE_{mathcal{T}}[Y|X^0]\
&= X^0 (X^TX)^{-1}X^TXbeta
end{align*}

The second equality holds assuming that the covariates $X$ are fixed by design, so the only thing that’s random with respect to the training set $mathcal{T}$ is $Y$, correct?