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You'll probably want to wait until spring for this. Nonetheless, the following resource will help you:
http://lee.ifas.ufl.edu/AgNatRes/Pubs/Pruning_Avocado_trees.pdf

We would not recommend pruning an avocado that much, especially when it is that small. Right now, you can take maybe 2-3 inches off, but any more than that will likely kill it. You don't want to prune more than 1/3 of a plant in a year.

To prune it, just use a sharp pair of scissors to snip off the part of the plant you wish to remove.

Yes, choose a few of the strongest shoots and cut away the rest. These will grow into the new trunks.

You can try to root the cuttings, but they are very difficult to root, even under the best circumstances.

This article will help:
https://www.gardeningknowhow.com/edible/fruits/avocado/avocado-tree-growing.htm

It does sound like a watering issue. This article may help: https://www.gardeningknowhow.com/houseplants/hpgen/how-to-water-a-plant.htm

In addition, you may want to have your soil tested to see if it is lacking any nutrients and treat as necessary.

This article should help: https://www.gardeningknowhow.com/special/children/sprouting-avocado-pits-how-to-root-an-avocado-seed.htm