The production of two metrics is a metric also. It's googled easy. But what's about a sum? As I can see sum is metric, as the triangle inequality of metric sum is the consequence of the inequality feature and two other axioms looks obvious. Is it correct?
In short, yes. Your sketched argument is correct for the triangle inequality (we're just adding two valid inequalities to get a third). One does also need to remark that the sum of two numbers $\ge 0$ can only be $0$ if both summands are $0$ (and then we apply the axiom for the two constituent metrics after that).