Cats and dogs primarily. But see the quote from St. Francis of Assisi at #10 for a general statement of our responsibility.
How are they like "eternal children"?
When we take on a pet, we stunt its growth as a "feral" animal. They don't ever grow up, since they don't learn to hunt or forage. They are children for their entire lives.
What is your basis for those statements? Personal opinions? Tradition? Scripture?
Yes.
Yeah, I saw that, and it left me with the same question. "Any of God's creatures" would seem to include chipmunks, mice, rats, snakes, spiders, ... So, I also have to wonder how you feel about mousetraps, flyswatters, etc. I suppose it's conceivable that my dispensing a quick death to them is showing them "compassion and pity." But, is it okay to put out D-CON to control mice and/or rats? How would you feel about my drowning them?
They are children for their entire lives.
You believe they're "eternal"? If so, on what do you base such a belief? Please be specific.