If I change my cassette, should I also change the chain? If I change my cassette to a new one, should I also change the chain?

I measured the chain with a wear gauge and it's ok.

Just to emphasize the obvious: I'm changing the cassette to one of exactly the same type.
Depends on how worn the chain is. If it's relatively new go ahead and use it. If it's at or approaching 0.5% stretch, just get a new one.