I think the others pretty much covered it already but diatomaceous earth is dead sea critters. When viewed under a microscope you'll find they come in all kinds of ninja star shapes. It kills by cutting through the protective layers of insects and they end up dying of dehydration. As you already know it's a super fine powder and can easily be washed away with the garden hose, rain, peeing dogs, etc., so in order for it to kill things it has to be present for bugs to come in contact with. The general rule is to apply it for sure after every rain, and if you're spreading around the ground or base of your plants you'd need to do it all over again after watering.
Ants hate lemons and oranges. If you spray lemon juice or blend some orange peels with some water in the blender and spray it around (or better yet pour it down the ant hill if you can locate it) they'll stay away. They don't like pepper either (hot pepper...like cayenne powder) Boric acid mixed with sugar or peanut butter will also kill them but it will also kill any other bug that eats it and probably isn't a good idea if you have pets or toddlers or anyone else who might end up eating a pile of peanut butter off the ground.
I don't know how many plants you're dealing with but I've had good luck blasting the aphids off with the garden hose daily until they're gone. Soapy water will suffocate them. Just use a little dish soap (not the antibacterial kind) in a spray bottle full of water and spray them.
