I have used all of the above. Living in Florida, palmetto bugs and ants are a given. And sometimes lizards. All of which make certain birds turn into screaming velociraptors.
We used an exterminator until we knew what he was using then did it ourselves. The gel works well on window ledges. I don't do the traps because the dog would probably get ahold of them. He was putting DE inside and outside so now, that's what we do. If you find a trail of ants, put some down. They'll be gone in no time. The bug man told me they carry it back to the nest, a know-it-all friend told me that's not true. Use your judgment.
AIL is used in the bird room and around the perimeter. You can also spray it directly on birds, I don't think I've ever done that.
DE is used everywhere, the thing about inhaling it is -- just don't inhale it. And don't make it into a cloud. Use an applicator that 'spits' it right where you want it. It's safe for birds and other pets, and you don't have to worry. We put it in the tray beneath the grates (for the birds that have grates). Some people suggest putting it in bird food because it deworms them. The cavy people say that too. *I don't use it for deworming.*