What’s a safe way to clean tough messes (urine, etc.) from carpets?

There are a number of ways to clean carpets without dangerous chemicals. Blotting is a good way to clean up stains quickly, Blot with white paper towels or clean fabric; stand on the absorbents so they penetrate more deeply into the stain, then remove. Don't rub the carpet with the paper towel or fabric; just repeat using clean absorbent material.

Scoop as much of the stain as possible onto a spoon or appropriate implement.

Nothing works better than baking soda to remove odors from a carpet; it literally neutralizes them on the spot. Sprinkle baking soda over the stains and leave it on overnight. You may need a box or two to really do the trick. To avoid clogging up your vacuum cleaner, sweep off as much baking soda as you can, then vacuum up the rest. For a urine neutralizer use borax or baking soda or spray of club soda (do not rinse; soak up moisture with a sponge if necessary).