Why would you care how non-Christians think? Are we not 'evil' and/or 'wrong' in your world view? Or are you hope to use it to try and win converts?
I agree the study of philosophers can help with critical thinking. I enjoy Thomas Aquinas' writings. I also enjoy Hobbs, Locke, Rousseau and Voltaire. From the East I like Lao Tzu and Chuang Tzu as well as Sun Tzu. Not quite philosophy perhaps but I also like Miyamoto Musashi along with other writings of the Sengoku Period Samurai.
Great questions, Leah!
In regard to your first question, I'd just say that Christians should care about how everyone thinks, including other Christians. The 'why' of it is because since we are supposed to love other human beings--even those who swear opposition to us as enemies--we therefore commissioned to do our best to understand others, reach out to them, and if possible, help them. However, if we (Christians) don't gain insight into why people may be acting or thinking as they do, we have very little chance of helping them. It would be like a person wanting to become a clinical psychologist without actually doing the studying and work required to actually become a clinical psychologist. On a more practical level, the study of how 'non-Christians' think can help Christians understand where other may be coming from.
As to your second and third questions, Leah, I'd say that non-Christians are human beings just like everyone else, and from a Biblical perspective, we are ALL sinners, all capable of doing evil. Christians are supposed to develop empathy, compassion and love for others, and it isn't just to win converts. Remember, Jesus told His disciples to give without expecting anything in return.
Like you, I've read some Lao-Tzu, Chuang Tzu (for my Eastern Philosophy class), some Hobbs, Locke, and Rousseau, as well as others staple figures like Descartes, Hume, Kant, and Hegel. But for me, it is the modern philosophers that I enjoyed the most: Russell, Carnap, Quine, Sellers, Dewey, Wittgenstein, etc. (And I guess I'll have to check out SunTzu sometime--that might apply in some ways too. )
Yes, all of this can help, although a good dose of logic and argumentation can help to some extent as well. I wish more Christian would avail themselves of these disciplines, along with more knowledge of history and ethics. Unfortunately, since we live in a fallen world, I can't expect too much from my fellow human beings.
Peace, Leah