What exactly is purgatory?
Purgatory is a state where the soul is purified and made ready for heaven. Remember, Scripture says that no one who is not perfectly pure will enter Heaven. Most people have some attachment to sin, even if they die in the state of grace. Here’s an example of “attachment to sin.” A young man and woman are at the altar, getting married. It comes time for the “I do’s” and the young man immediately responds with “I do.” The young woman, however, is distracted, looking out into space like she’s not there. After a gentle nudge, she comes back to the present situation and says, “Oh, yeah. I do.” Later, the new husband asks her what was up then and she says, “Oh, I’m your wife forever, and I love you, but at that moment, I just couldn’t help thinking about my previous boy friend and how great he and I were together and how much I love him. Well, that didn’t work out, so I’m your wife now and I’ll love you forever.” How would that young man feel? Well, that’s how God feels when we have some internal attachment to sin (usually some pet sin of ours). He wants our total loyalty and faithfulness, inside and out.
Is it the Catholic belief that most properly baptized (Trinitarian formula by immersion) Christians go to Purgatory if they are not Catholic? The Church usually refrains from saying who went to hell or purgatory. It’s not about being Catholic or not, it’s about dying in the state of grace and having some attachment to sin, even forgiven sin. (See above explaination of “attachment to sin.”
How can someone else’s prayers change the eternal fate of someone? They can’t. Once one dies, one is immediately judged by Jesus Christ and sent to hell or heaven (with a possible side trip to Purgatory on the way to Heaven). Final destinations (eternal fate) are either heaven or hell.
Wouldn’t they be responsible for themselves(ie their faith or works as the case may be) despite the best wishes or prayers of others? That’s one of the big difference between Catholics and non-Catholic Christians. We believe that we are members of the Mystical Body of Christ, the Communion of Saints. We are our brother’s keeper, so to speak. It’s not “Me and God” like it seems to be in Protestantism, but “We and God.” The Communion of Saints consist of those on earth who have been baptized, those in Purgatory, and those in Heaven. Jesus said that He is the Vine and we are the branches. Branches don’t get separated by earthly death. In fact, if we make it to heaven, we are even more united with Christ, the Vine, than on earth.
On a side note, what do Catholics believe will send a person to hell? Please be specific and brief if possible. Dying with one or more unrepented mortal sins. (Is that brief enough?

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