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You can cut back on watering or allow the soil to dry slightly between waterings.

You can move the plant to an area that better air circulation.

Irish Moss may be a good choice.

https://www.gardeningknowhow.com/ornamental/groundcover/irish-moss/growing-irish-moss-plants.htm

Do not store in plastic bag. Lay out the moss in a shady area and keep slightly damp. Use rain water or distilled water for any moisture needed.

Conditioned water can kill the moss.

Squirrels dig wherever the soil has been worked and is super easy for them to dig in. You can try ground hot pepper, predator urine or other DYI recipes found on the web. But they must be reapplied after watering/rain. Your best bet may be to cover the area with hardware cloth using landscape fabric pins to hold it down. Moth balls are said to deter squirrels but are toxic to animals, including children, who eat them. A more exotic approach is to get a motion-activated lawn sprinkler. Would that stop a determined squirrel?

You can do this just about anytime, although it won't do near as well in the cold. Spring time would be your best bet. Here is an article that will help you to establish a moss lawn:https://www.gardeningknowhow.com/lawn-care/lawn-substitutes/moss-lawn/how-to-grow-a-moss-lawn.htm

The most common moss is, actually, called "sidewalk moss"! Bryum caespiticium, or sidewalk moss, can be easily propagated on most surfaces. This link will take you to a collection of articles that will help: https://www.gardeningknowhow.com/ornamental/foliage/moss/