The form of the residual vector in the multiple linear regression model

I have 2 questions about multiple linear regression model.

Does the residual vector in the multiple linear regression model have the form $$y-(X(X^T X)^{-1} X^T)^2 y$$?

And, is it true that the sample distribution for a linear combination of model parameters in the multiple linear regression model is used in deriving the prediction interval for a future observation $$x_0$$?

Yes, the residual vector is correct but the expression can be simplified because: $$(X(X^TX)^{-1}X^T)^2y=X(X^TX)^{-1}(X^TX)(X^TX)^{-1}Xy=X(X^TX)^{-1}X^Ty=X\hat{\beta}=\hat{y}$$ and residuals are $$r=y-\hat{y}$$. You don't have to square the term in the parentheses.
Since $$\hat y_0=\hat{\beta}x_0$$, prediction interval will use sampling distribution of parameter estimates.