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I would not plant a young seedling directly into compost. I would mix with some inert ingredient such as coconut husks, or peat moss. I would wait to transplant until it has a vigorous, branching root system. Then I would still say you want to mix in other ingredients that just one. Usually compost that lack balance will cause many deficiencies. If you are going to do this just to see the deficiencies, you will still want to have well composted materials. Here is an article about composting: https://www.gardeningknowhow.com/composting/basics/ultimate-beginners-guide-composting.htm

And here is out collection of tomato articles: https://www.gardeningknowhow.com/edible/vegetables/tomato/

Unfortunately, it will be hard to give any advice without a photo. Tomato stalks have a normal bumpy growth that will turn into roots if allowed to touch the ground. This may be what you are referring to, although I couldn't say for certain.

As for not bearing fruit... There are many reasons for this. This article will help: https://www.gardeningknowhow.com/edible/vegetables/tomato/tomato-blossoms-no-tomatoes.htm

It can be done, but most of the time it will rot, or attract insects. Sometimes its not always a good idea to do something just because it is possible.

The best thing to do will be to scrape a few of the seeds from a tomato when you slice it and then plant one or two seeds. There will be a problem when more than one tomato seed sprouts. They don't tend to grow well with multiple plants together.

In short, this is a poor method for starting tomato plants. It is nothing that you did wrong, but the method its self is faulty.

Beware of "lifehacks". Some can be useful, but I find that many of them just create more problems than they solve.

I recommend not thinking in terms of a limited number of individual plant nutrients, but use a complete organic fertilizer that supplies as much of the full range of mineral and biological amendments as possible.

Agricola's 4-8-4 is the best I know of:
http://www.soilminerals.com/Agricolas4-8-4_MainPage.htm
dr-earth-home-grown-tomato-vegetable-herb-fertilizer-p-9493.html
Next best may be Dr Earth and Down To Earth All Purpose or vegetable blends:
https://www.domyown.com/dr-earth-organic-all-purpose-fertilizer-p-9485.html
https://www.domyown.com/dr-earth-home-grown-tomato-vegetable-herb-fertilizer-p-9493.html?sub_id=9492

If this is blight, and you are certain that it is not tobacco mosaic virus, then it can be cured. Just add wettable sulfur and dolomitic lime to the area, and wait a few weeks. If this is TMV then you will have to replace the soil and bed materials, as it is not able to be cured.

If you can send me some pictures, I will be able to better assist you, as it could also be something really simple to cure.