You need Kim to look in here.
Looks like I'm late to the party. But yes, voltage ratings are a safe maximum, no issue with using a higher one, if it physically fits (higher voltage ratings tend to mean physically larger capacitors). Using a lower voltage rating is a recipe for magic smoke release.
Class X and Y mean they're safety rated (to withstand voltage surges) for mains applications. X-rated capacitors usually fail short-circuit, which is appropriate when they're between live and neutral (there's presumably a fuse that can blow to prevent fire). Y-rated means they're designed to fail open circuit, which is necessary to prevent electric shock risk when they're connected to earth.
Flux capacitors are rated in Jiggawatts, rather than the usual Farads.