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Yes, seedlings can be started in a soil-free mix. An alternative to what you've proposed is a commercial mix that is specifically intended for starting seedlings.
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soilless-growing-mediums.htm

Liquid fish fertilizer is a good choice for seedlings and should be applied starting when the seedlings develop their first true leaf (besides the initial two cotyledons or seed leaves).

Neem oil may not be safe for young seedlings, so prevention is the best option for diseases and insects. Check the moisture on the tray often: keep the soil evenly moist but not soggy to prevent mold. If you see any mold, decrease watering.

Appreciate any ideas please

The most important thing would be to clean out the tank thoroughly to make sure no oil residue remains on the surface. If the tank is plastic, it may not be possible to clean it thoroughly enough, since the plastic may have absorbed oil. In any case, you should line the tank with a plastic raised bed liner before adding the soil, to prevent the plants from absorbing any toxins.

Other than this, look over the raised bed articles on the site, such as the following, to learn about soil options, etc:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/best-soil-raised-garden-beds.htm

There will be no harm in growing with shoeboxes! The only thing that I would recommend will be wrapping, or painting the outside so that light does not burn the roots.

Here are our container gardening archives. These will container quite a bit of useful information on container gardening: https://www.gardeningknowhow.com/category/special/containers

I'm guessing the plants do not get enough sun. Most vegetables require full sun which is 6 to 8 hours. Here are more general gardening tips that may help.

https://www.gardeningknowhow.com/special/urban/balcony-vegetable-garden.htm

https://www.gardeningknowhow.com/edible/vegetables/vgen/quick-guide-to-container-vegetable-gardening.htm

https://www.gardeningknowhow.com/special/containers/best-container-vegetables.htm

https://www.gardeningknowhow.com/edible/herbs/hgen/herb-plants-growing-in-one-pot.htm

https://www.gardeningknowhow.com/edible/herbs/hgen/growing-herbs-in-containers.htm

It can be fine to grow in a plastic swimming pool, as long as you drill plenty of drainage holes. Otherwise, the soil would become too saturated during rains.

What you can grow in there will depend on its depth. As long as it is 12 inches or deeper, you can grow pretty much anything in there, except for large trees.