Is moissanite a good diamond alternative here?

Yes. Moissanite is harder than cubic zirconia and keeps its brilliance longer, though it has more rainbow fire than a diamond. Most people cannot tell the difference at arm's length.

Will the white gold vermeil fade?

Vermeil is a thick gold plating over silver, so it lasts longer than thin plating but still wears with sweat, perfume and friction. Take it off for showers and workouts to extend its life.

Is a 6mm tennis necklace too wide for everyday wear?

6mm is a bold, statement width. It suits evenings, layering or a flashy everyday look, but people who want something subtle usually prefer 2 to 4mm.

Yes. GLD lists it as unisex, and the curb and Byzantine chain styling and 6mm width work on both men and women.

How should I care for it?

Wipe it with a soft cloth after wear and store it dry and separate from other jewelry. Avoid chlorine and harsh cleaners, which speed up plating wear.