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Adding the neem oil is fine anytime and should not interfere with the fertilizers. Be sure to spray the neem on the leaves as this is most effective.

It sounds like you have whiteflies. The neem should take care of them, but here is some more info on them, just in case you wanted to know more: https://www.gardeningknowhow.com/plant-problems/pests/insects/whitefly-control.htm

I would fill the containers with something that is well draining such as peat moss mixed with potting soil that has been amended with compost or well-rotted manure.

Yes, either of these options will work. You can even use Styrofoam peanuts instead of rocks in the bottom of pots if you are worried that weight may be a problem.

The following articles should be of some help to you:
https://www.gardeningknowhow.com/special/containers/upside-down-gardening.htm
https://www.gardeningknowhow.com/edible/vegetables/vgen/vegetables-grown-upside-down.htm

It can, but you need to be aware of 2 things. First, the nutrients in the soil will have been depleted, so you will need to replace those with either adding fertilizer or compost. Second, old potting soil can harbor pests and disease, so it is best to sterilize it before reusing. Here is an article on sterilizing soil:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/sterilizing-soil.htm

For plants requiring little sun, this article should help: https://www.gardeningknowhow.com/edible/vegetables/vgen/shade-vegetables.htm

I think containers may be the better way to go since you have more flexibility and can move them to a sunnier location. Here is more information: https://www.gardeningknowhow.com/special/containers/designing-your-container-vegetable-garden.htm

As for the run off from the rain, why not get a rain barrel. Rain barrels can be attached to the downspout and will collect rainwater that would normally run into the yard. You could then use this to water your plants.