Spraying with most every pesticide not only kills off the insect pest, for a time, but kills off any beneficial insects as well and those can include predators of the pests you may have. Therefore spraying is not the real solution to the problme. Spraying may work short term but it does much ecologoical harm long term.
Plants growing in a good, healthy soil that are strong and healthy are less desireable to insect pests, so the first thing to look at is your soil.
When was the last good reliable soil test done?
What is your soils pH? How much organic matter is in that soil?
How well does that soil drain?
How well does that soil retain mositure?
What is the life in that soil like?

I would agree with kimmsr about pesticides killing predatory insects. Sunflowers are generally pretty hardy and although not pest free, generally not super attractive bug bait.
The ants are probably going for the honeydew produced by the aphids. The aphids are probably there due to the lack of predators. You might try backing off the pesticides and, in the event of aphid infestation, just wash them off with a strong stream of water.
With no pesticides the predators will eventually re-emerge and likely keep the aphids under control. Don't know what to say about the moths, never had to deal with them - geez, embarrassing as I live in the Sunflower State!
hortster

Something worth mentioning is that sunflowers can tend to be attractive to a wide number of pests....no matter how healthy the soil is. Aphids, spider mites, beetles, weevils, and caterpillars all seem to love these plants.

True caterpillars can be kept under control with Bt-k (Bacillus thuringiensis- Kurstaki, a biological control that targets plant eating caterpillars (and only caterpillars). It must be sprayed on the plant while the larvae are still very small, but will stop them from feeding as they begin to die off.

Bt-k is not harmful to any other insect, nor will it harm birds (or others) who come and eat the affected caterpillars. Bt-k comes in liquid and dust formulations, but I strongly suggest that you located some of the liquid.

Be sure that you are purchasing the Bt strain for caterpillars and not the stuff for mosquitoes or fungus gnats. Kurstaki is what you want.

Something worth mentioning to you is that the use of some pesticides actually creates an environment in which pests can multiply like crazy. Horster and kimmsr were hinting at that. It's a known fact that the more you use broad spectrum insecticides, the more you will HAVE to use them.

Bt doesn't fit into that category.

rhizo_1 - I think you may be right.

Could DE dusted on the leaves help against caterpillars?

Here is a link that might be useful: Sunflower Pests