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Composting plant materials is always preferable to trashing them. This article will introduce you to the basicshttps://www.gardeningknowhow.com/composting/basics/composting-basics.htm of composting:

I think this depends more on the type of charcoal used. If it's natural wood charcoal then it's okay to put the ashes in your compost. If it is the type that has additives (chemicals), including lighter fluid, then I wouldn't use it in compost. Check the ingredients and the label. Also, be sure to let it sit for several months to age before using on plants when putting the ashes in the compost.

To prevent smells in the compost pile, you need a good balance of greens and browns. This article should help: https://www.gardeningknowhow.com/composting/ingredients/browns-greens-compost.htm This article will help fix the problem: https://www.gardeningknowhow.com/composting/basics/compost-smells-bad.htm

There really is not any benefit to removing the thin lining from the eggshells. There is a lot to be gained by smashing the eggshells up into as fine a pieces as possible though. It can take a very long time for them to break down into truly giving the full benefits to the compost pile and the gardens if left in large pieces. Breaking them down into very small pieces speeds up the break down process and gives you their full benefits far far quicker. Here is more info: https://www.gardeningknowhow.com/composting/ingredients/eggshells-in-the-garden.htm

This article should help you with that: https://www.gardeningknowhow.com/composting/ingredients/using-sawdust-in-your-compost-pile.htm If the termites no longer have anything to feed on, they will likely die or move elsewhere. The heat from the compost pile may also kill them, so this shouldn't be a problem. If they do survive, it is unlikely they would cause any damage to your plants, as they typically only target unhealthy trees.

Ammonia smells from compost usually mean too many greens and not enough brown material in the pile. Here is more information: https://www.gardeningknowhow.com/composting/basics/compost-smells-bad.htm, https://www.gardeningknowhow.com/composting/ingredients/browns-greens-compost.htm