My mom is the gardener in our family, so I'm getting a lot of this info secondhand, but she's always careful to make sure anything she uses wouldn't harm my girls or the native bees. What I always tell her to remember is to ask yourself how the insecticide kills the insects. The bees are only visiting flowers, so if whatever you are doing doesn't affect the flowers, it should be pretty safe. For example, something like neem oil requires the insects to eat the plant it's been sprayed with. As long as you don't spray open flowers, that's going to have essentially no impact on a hive. Something like dishwashing liquid is meant to drown small insects, so as long as the liquid is just sprayed on the plants, not much impact there. DE is generally okay from what I've heard, but be careful about using it in a dearth, as bees have been known to collect other dusts when there is no pollen. I would steer clear of strong broad spectrum insecticides like Sevin, and always be careful about spraying open blooms and plants treated with systemic insecticides.
Also, in my experience, chickens IN a garden is a romantic fantasy that does not reflect reality. I can see having them in a fallow garden, but in an active garden with plants, that would be a disaster. We tried letting our chickens in our garden the first year, and they ate and/or destroyed everything.