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I disagree with virtually everything about Westboro. They call themselves Baptists, but they do not hold to historic Baptist, or Biblical, beliefs.
I don't want to get too theologically technical, but the root of their problem is that they are very strong Calvinists. In other words, they believe that God chose, in eternity past, certain people to go to Heaven and the rest to go to Hell. They, of course, are chosen to go to Heaven (they think some other people are too, not just them). Since a person's salvation is pre-determined, and they have no say in it, the Westboro folks do not believe in telling people that they need to get saved, or inviting people to receive Christ, or going "soul-winning" in the traditional sense. After all, how can you tell someone that God loves them and wants to save them if you believe that God has already predestined them to Hell? That wouldn't make any sense. Thus, what they do, instead of inviting people to Christ, is pronounce God's judgment on the lost. They believe that God only loves those who are predestined to Heaven, and that He hates those who he predestined to Hell, so that's why they hold up the "God hates _____" signs. They believe that violent deaths like these, as well as the deaths of American soldiers, are signs of God's judgment upon America for its sins, most notable the sin of homosexuality.
The bottom line for Westboro theology is this: God hates the vast majority of humanity. He predestined most people to go to Hell, and thus Christ never died for those people, just for those that God chose to go to Heaven. Since God hates them, it pleases Him to see them suffer and die. Since salvation is completely up to God, then the only real message they can "preach" is that God's judgment is coming, and is being revealed through the deaths of these children.
There are numerous other problems with the Westboroites. They exhibit very little personal holiness. They live for lawsuits, which is probably how they fund their "ministry" (a number of the pastor's family members are lawyers, and he may have some expertise in that area as well). Obviously, their grasp of Biblical truth is minimal.
I do not believe that the deaths of these children is God's judgment against sin any more than the death of anyone else is God's judgment against sin. Sometimes God judges sin by taking away life - the Bible has plenty of examples of that. However, most people die simply because sin is in the world. The Bible says,
"Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned" (Romans 5:12).
Thus, all death is ultimately a result of sin, but not necessarily the direct judgment of God for any particular sin.
Did God want it to happen? That's a tough question to answer, because I don't know the mind of God. We do know from Scripture that sometimes God allows Satan to do what he wishes. However, just because God allows something to happen doesn't mean that He wants it to happen. For example, He doesn't want me to sin, but He doesn't kill me to keep me from sinning. We know that God could have stopped the tragedy, but in His infinite wisdom chose not to do so. Why? I don't know, but here is one possibility: I would say that there is a pretty good chance that most of the children who died were being raised by unsaved parents (I am basing that statement on the fact that most people are unsaved, so the odds would be good that most of those families would be unsaved - I obviously know nothing of the spiritual condition of those families in particular). Since most of those children were so young, many had probably not reached an age where they could choose to accept or reject Christ, and that being the case they would have gone to Heaven when they were killed that day (II Sam. 12:23). Had they reached maturity under the influence of unsaved parents, who knows where they might have wound up in eternity? If the children were from the families of believers, we know that the Bible teaches that "all things work together for good to them that love God" (Romans 8:28), and God never tells a lie (Titus 1:2).
Finally, remember that God knows everything, and does everything absolutely correctly, and thus does not necessarily do things the way that we finite, simple, sinful humans think that He should. Isaiah 55:8-9 is a good passage to remember:
"For my thoughts are not your thoughts, neither are your ways my ways, saith the LORD. For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts."