"…I just don't think it's fair that my far-from-perfect soul can go to heaven where people like, for example, Mother Teresa went. I have not dedicated my life to others like she did."
--> This is why there are different levels of heaven, purgatory, and hell. Each person goes to their own level. (You must also consider that heaven and nirvana are totally different, which gives people even more levels to go to after death.)
"I feel like the rich man who asked Jesus what to do to be received in heaven…"
--> Every rich man will go to his appropriate level of heaven, purgatory, or hell, depending on his level of compassion and how much (or little) he helped his fellow man.
"If you declare with your mouth, "Jesus is Lord," and believe in your heart that God raised him from the dead, you will be saved."
--> What about people who don't do this? Do you think everyone who doesn't do this will go to hell? Tad, have your parents heard of Jesus but chosen not to follow Jesus? Do you believe they will go to hell because of this? Mahatma Gandhi, one of the greatest men of the last century, chose not to follow Jesus. Do you think he will go to hell because of this?
"…what about babies, small children, people with cognitive impairment (e.g., down syndrome, severe autism, Alzheimer disease...), and the people who have never heard of Jesus, or never read the Bible before they died??"
--> The twin ideas of reincarnation and enlightenment remove all of the unfairness of these situations.
"I think I still want to believe animals can reincarnate into humans (but not the other way around, hopefully). Do you know of any theosophists who think that way too?
I know it's silly but I sometimes feel like I love my cat like a son and fantasize him eventually being born into a human, and the way I take care of him is actually helping to shape his soul and his personality and that he'd be born into a kind human being…"
--> All Theosophists believe that your cat will become a human being. But this will happen on the next ‘earth’, not on this earth. You have nothing to worry about, your cat will have many, many incarnations as a human.
You are correct, humans do not reincarnate as animals.