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Most insecticides have a time period that they break down in. Simply waiting that time period will cause it to go away. The time period varies depending on the type you used. You can also soak the vegetables in water for a few hours and discard the water.

In the future, you may want to consider using a human consumable insecticide. We recommend neem oil. It will kill bad bugs but has no adverse effects on people and pets. Here is more information: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

I like neem oil for pests like whitefly. It is safe for people, pets and beneficial bugs but very effective on pests that attack plants. I have even seen it work on vine borers and these are near impossible to kill with pesticide. Here is more information:
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Broccoli will not do well upside down, but that being said, one of the nice things about using the upside down method is that you can make use of both ends of the bucket. The broccoli can certainly be planted in the soil on the top of the bucket.

Any plant that is considered a cool weather vegetable will do ok in the shade during hot weather. Cole crops, like broccoli, cabbage and greens, are some, as well as lettuce and peas.

No, not all broccoli has these worms. If you are buying commercially grown broccoli, you should not find any worms because they use pesticides to stop them. Most organically grown or home grown broccoli will not have caterpillars or worms, but some will slip through. If you are eating organic or home grown broccoli, it does not hurt to soak the heads, but you will find 99 out of 100 times they have no pests.

Just cut out the head. You may actually get side shoots growing afterwards that can also be harvested.