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If the cardboard is shredded pretty well and fine and is free from tape, staples, ink, plastic, and Styrofoam, yes--you can put in your garden. At that stage though it will serve more as mulch rather than compost. Some people mix it in with the garden soil.

We only have citrus trees and grow no vegetables because of heat, etc. Still harmful?

Ficus leaves should be fine to use in a compost pile, especially a "hot" compost. The only exception is if toxic pesticides have been used on the plant that you wouldn't want to use on your edible garden.

Snail eggs and snails can appear in compost bins. Based on the description, you could also have fungi and you could be seeing the fruiting bodies, but it's hard to tell without a photo. Fungi in compost would be nothing to worry about.

You don't want to add live snail eggs to your garden, but if you can get the compost to heat up, that will kill the eggs. When the compost is finished, it won't have much for snails to feed on anyway.
https://www.gardeningknowhow.com/composting/basics/heating-up-compost-pile.htm
If there are just a few snail eggs, you could always scrape them off the compost too.

Here is more information about identifying snail eggs:
https://www.gardeningknowhow.com/plant-problems/pests/insects/slug-snail-eggs.htm

This article has great information on trench composting.

https://www.gardeningknowhow.com/composting/basics/trench-composting.htm

I good ratio is 1 cup tea to 1 gallon of water.