My main commuter is a freebie '97 Fuji Odessa that has Alloy Weinmann 26x1.5 rims, but I've noticed that after using my brakes a lot (e.g. after riding in the rain) I have readjust and tighten my v-brakes. I replaced the pads but the same thing happened. After taking the brake pads off there are always aluminum bits in them. It seems like my pads are wearing down a lot more rapidly than they should be. Does this mean the rims are worn out?

It means your brake pads aren't very good. Buy some new decent ones.

A rim is worn out when the braking surface is noticeably concave. It's best to replace them before they crack and fail.

Feel the brakeing surface of your rims with your finger. If they feel noticeably concave, you're ready for new ones.

Also regularly pick out the sand and alloy bits from your brake-pads. Leaving them there turns your brakes into sanding-pads and wears out the rims faster.