1. Bacon (or other greasy foods). How to cope with food items that require blotting? Or are cooked in the microwave inside paper towels?
Answer: Have a very large white cotton cloth that is for food-oil blotting only. Cook bacon
-- or fry tortillas or fries or tofu -- on the stove in a pan (natch). Then lay out your special cloth on the counter or fold on a plate, and blot the grease the same way you would with paper towels. Hand wash the cloth afterward with a degreasing soap like Dr. Bronner's. It's definitely more work, but you're no longer wasting 10 paper towels to make breakfast.
2. Disgusting messes such as a squished bug or cat barf (or worse). If you've given up paper towels, how do you clean up the occasional extremely gross thing? You would never want to use the cloth again.
Answer: Collect and store thin pieces of cardboard or subscription coupons that fall out of magazines to deal with solid messes. Slide the cardboard piece underneath slowly, then lift. Place on compost pile (if applicable) or throw out. For less solid messes, rip or cut off a piece of your oldest, yuckiest rag to clean up the gooeyness, then throw it on your compost pile (if you've got one) or throw out.
Note: I am a cold-turkey, out-of-sight-out-of-mind person, so I don't even keep paper towels in the house anymore -- I would end up using them all the time again! But I could totally see a more strong-willed person being able to have a roll or two around for oil-blotting or super-yuck stuff.