Sure, so on purgatory. This will be quick and dirty and peripheral. This is hardly comprehensive, but gives you an idea.

We follow St. John's teaching that there are mortal and venial sins.

ἐάν τις ἴδῃ τὸν ἀδελφὸν αὐτοῦ ἁμαρτάνοντα ἁμαρτίαν μὴ πρὸς θάνατον, αἰτήσει, καὶ δώσει αὐτῷ ζωήν, τοῖς ἁμαρτάνουσιν μὴ πρὸς θάνατον. ἔστιν ἁμαρτία πρὸς θάνατον· οὐ περὶ ἐκείνης λέγω ἵνα ἐρωτήσῃ.

If anyone sees his brother committing a sin not leading to death, he will ask (God), and God will give him - to those who commit sins not leading unto death. There is sin which leads to death. About this kind I do not say one ought to pray. (1 Jn 5.16)

So we have a division of sins. If someone dies in God's grace and friendship but has venial sins still on their soul, the eternal consequences are certainly paid for in the Cross, but the person himself still has the habits, the loves of things other than God, or whatever, still on their soul. Since nothing but the perfectly holy can be in Heaven with God, purgatory is a state where you can think of those wrinkles in one's soul, created by the venial sins, are ironed out. It's the fulfillment of Christ's promise to sanctify you.

Think about it this way: God respects the fact that you're a being with a will. If you died right now, could you say that you love God completely? I love God quite a lot, but I doubt my love for Him is perfect. Purgatory is a state for those who have died in God's grace, but may still have those old loves (which interfere with the love of God) clinging to their soul. They have to be purged of them, sanctified totally (all due to God's grace), before they can enter Heaven and actually enjoy it. If God didn't respect who you are, He'd just snap His fingers and change you the instant you die - but that seems sort of absurd. What in the heck is the point of living your life if all you have to do is die and then suddenly you're a different person who now, somehow, completely loves God?