In my area there are very few flea beetles in fall compared to spring. If you plant some arugula and mustard, the ones that are around will go after that instead of your broccoli.

I probably would have assumed the same thing. I think they are active during the day, but I don't really see many flea beetles here at all, so I'm just guessing about that. There are many different species of flea beetles and different ones are active on different kinds of plants. How much to worry about the damage they are doing would just depend on various factors including whether they are damaging a plant enough that its growth is stunted? Or, does the plant continue to grow and produce just fine? Most of the time, the damage is just a matter of appearance, except that when flea beetles hit seedlings or very small plants early in the season, they can damage them enough to kill them. Your description of the leaf damage sounds exactly like the damage that flea beetles do. Flea beetles are not always easy to see. I suspect they are smart enough to hide when people are around. If you can walk quietly into the garden and approach plants with flea beetles on them within them becoming aware you're getting close to them, often it startles the flea beetles and they sort of hop up into the air as they attempt to flee (pun intended) once they are aware of your presence. The thing that puzzles me about you seeing flea beetles this late in the summer is that, at least in my part of the state, they are mostly a late winter/early spring pest. Of course, this has not been a normal weather year and some kinds of flea beetles can remain active at least into June...which wasn't all that long ago. If y'all have had fairly abundant rain and cooler than average weather, even prior to this week's cold front, then I imagine the flea beetles still are active. In our climate, we can have several generations of flea beetles each year. There are some foliar diseases whose damage also could give your plant foliage that buckshot appearance, so I do think it is important to try to determine if you are seeing flea beetle damage or a foliar disease. Some of the bacterial diseases that cause foliage to have specks or spots often do progress to tissue necrosis where the tissue within the specks or spots does die and fall out, leaving a bunch of holes that can mimic flea beetle damage. Normally, though, with those bacterial diseases you'll see dark specks or spots on the leaves before the necrosis occurs. So, with the damage you're seeing, do you see bacterial speck or bacterial spot on the foliage before the holes appear? Both of those can affect plants in the solanum family, which includes tomatoes and ground cherries.

Most everything I have seen about using Insecticidal Soap and Neem products is that they are not very effective against Flea Beetles, which presents a dilemma because starting with the least toxic method of control, pyrethrins, means starting with the most toxic as well and leaves little for fall back.

<Do eggplants need to be hand pollinated under cover? I thought they are self-pollinating, like peppers...> Have to? No just as you don't have to uncover them when they bloom for pollination. But it makes big difference in production just as it does with all other members of the family. They are also shielded from the wind when under cover and wind plays a role in pollination of eggplants, tomatoes, and peppers.