Yes, you can find this in Engelking's General Topology as Theorem 3.1.16 (for Hausdorff spaces but it works in general). See also Compactness and product spaces, Coll. Math., 7 (1959), 19--22 by S. Mrowka.

Yes, you can find this in Engelking's General Topology as Theorem 3.1.16 (for Hausdorff spaces but it works in general).

See also Compactness and product spaces, Coll. Math., 7 (1959), 19--22 by S. Mrowka.

See Problem 3.12.1 in Engelking's book for the following: if $X$ is not compact then there is a sequence $\langle F_\alpha:\alpha<\kappa\rangle$ of nonempty closed sets that is decreasing ($\alpha<\beta$ imlies $F_\alpha\supseteq F_\beta$) and has an empty intersection. In addition we assume that when $\alpha$ is a limit then $F_\alpha=\bigcap_{\beta<\alpha}F_\beta$. Then $\{\langle\alpha,x\rangle:x\in F_\alpha\}$ is a closed subset of $(\kappa+1)\times X$ (where $\kappa+1$ has the order topology and so is compact), whose projection along $X$ is the set $\kappa$ and hance not closed.

Yes, you can find this in Engelking's General Topology as Theorem 3.1.6 (for Hausdorff spaces but it works in general). See also Compactness and product spaces, Coll. Math., 7 (1959), 19--22 by S. Mrowka.

Yes, you can find this in Engelking's General Topology as Theorem 3.1.16 (for Hausdorff spaces but it works in general). See also Compactness and product spaces, Coll. Math., 7 (1959), 19--22 by S. Mrowka.