Can I replace them with a different set of icons?

Sure, just use the same numbers/filenames. Remember there are two styles/different sets of icons, so replace the style you like the least.

Also, after doing that, make a backup of the new set somewhere else on your hard drive, as installing a full version of the software will replace the icons with the default set again.

How do I know what the file names are? It's hard to tell, for example, "partly cloudy" from "mostly cloudy", etc.