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While you can deadhead if desired, this practice is not necessary. Your hibiscus plant will be fine left as is. The spent blooms will drop on their own.

I would cut back all the branches. Leaves will gradually come back on the branches that are not dead, but it will take a long time. Pruning often encourages plants to put out a new flush of growth. Here's some general information on pruning hibiscus. https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/tips-for-pruning-hibiscus-plants-when-to-prune-hibiscus.htm

Let's start with an article on growing hibiscus in containers: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/how-to-care-for-hibiscus-plants.htm
Yellowing and falling leaves, and no flowers, sound first like a watering problem -- is your plant getting enough water? Alternatively, roots that stay too wet could lead to root damage, and yellowing leaves.
After investigating watering, think about roots and soil. Is your plant severely root bound? That would prevent it from absorbing sufficient water. If you haven't repotted, you might try that, to give the roots more room, and to rejuvenate the soil. Here are a couple of article on this: https://www.gardeningknowhow.com/houseplants/hpgen/learn-more-about-repotting-houseplants.htm
https://www.gardeningknowhow.com/houseplants/hpgen/prune-roots.htm
Finally, consider minerals. Have you fertilized? If not, you need to - flowering takes a lot of energy. If you have fertilized often, you may have a salt build up. Running several gallons of water through the plant will rinse out the salt; repotting will also improve that situation.

Did the plugs not have growth when you got them? I think they should have. After two months you should have had, depending on the variety, from several inches to over a foot of growth. Hair-like growth sounds like mold - plant growth is going to be green. Here are a couple of articles on growing hibiscus: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/how-to-care-for-hibiscus-plants.htm
https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/wintering-hibiscus-indoors-winter-care-for-hibiscus.htm
Remember that growing hibiscus indoors, even just starting them, requires a minimum of several hours sunlight. And keep the soil damp, not wet. If there is water there when squeeze soil between your fingers, don't put more water on it.

I've never heard of any plant that was harmed by neem oil. However, most of the people recommending neem as a fungicide are the marketers themselves -- garden people generally use neem as an insecticide. Here are some other fungicide options: https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm