Cats often want to get warm, for example, toby being on a keyboard.

The example is describing a way that cats try to achieve this goal, so "by being" is more appropriate.

A nursery song says that when it's snowing more and more, a dog is willingly runnings around athe yard, andbut a cat curls up in Kkotatsu (a Japanese traditional heating appliance).

"Runs" sounds more appropriate because you are describing a situation in general. It also ensures that the structure is similar to the part that follows.
"But" is better since it highlights the contrast.

Surprisingly, I found a cat that used heating as humans wouldo.

This combination of tenses makes more sense to me.

You can find the photo googling with "火鉢 猫", which areis "hibachi and cats."

"Is" because you are referring to a phrase (singular), not the specific words in it.

Hibachi is also a (type of) heater.

It's a pot with ash and charcoal, and people hold outtheir hands over it.

The cat hangeld his forefoot onront paw on/over? the edge of the podt.

"Front/back paw" is more common.
"Hung" is the correct past participle of "hang," unless you are referring to the cause of death. That said, however, "held out" makes more sense to me. If you want to avoid repetition, I think "stretched" might also work.