Like you this issue is personal to me and I've spent some time thinking about it.
My current thoughts are pretty much summed up in two words, but extrapolated...
This is a sin-cursed and messed up world and in such a world messed up things happen without any [divine] reason or purpose, i.e. "**** happens".
I don't believe that God in any way causes or is responsible for the pain, suffering, and death that we experience in our lives. I mean, why is it that some people die in car crashes while others do not? Why do some people get cancer and others not? Of those who do, why do some survive and others die? The blunt answer is that that's just how it is in this world. Largely it's just chance and the natural consequences of our own actions or inactions.
To be fair, with regards to people being born with debilitating or abnormal conditions (which you brought up), I am not certain on this point. There is some Scripture that seems to imply that God is actively involved in the creation process of new human beings (Psalm 139:13-16). Is the Psalmist referring to the process itself which God instituted or something more (namely that God actually 'guides' the process) or something else entirely (like the 'conscious self' or soul)? I'll leave that you to because I'm not even sure I could guess. Suffice to say I don't believe that creating a young life with Down's Syndrome is in any way consistent with a loving and gracious God, assuming that He is in complete control of the process. Hence why I am more inclined to believe that it is the process that God instituted or the creation of a person's soul (that is, who they are or their conscious self/centre) that David is talking about.
All of that said, I do believe that God works through those painful times of intense suffering, disappointment, hardship, and death that we are bound to experience as a result of living here. It is often during those difficult times in our lives that we think about and become aware of just how much we need something more ... how much we need God. This may be reason why God tolerates and allows the evil, pain, suffering, and death in the world ... to show people their need for Him and that He has exactly what they're looking for and that He offers it freely if only they'd cry out to Jesus. God doesn't promise us a life free from pain, suffering, and death, but what He does promise is that when everything falls that we'd be held.
I believe that God can use our trying moments that life throws at us for our eventual good (Genesis 50:20, Romans 5:3-5, 8:28), or when no good can come of it (such as the death of a loved one) I believe that He comforts people. When my uncle died, my nan remarked that she felt God's strength and comfort give her the ability to deal with that difficult time of having to bury her oldest son. I don't believe that it was God's plans for my uncle to die, but in a fallen world, death is a natural part of it as is disease and cancer. God never promised that He'd heal him, but He did promise that He would go with my uncle through the valley of the shadow of death and give him the comfort and strength that he needed to finish the race well (Psalm 23:4) and I believe that He did just that.
The same thing I believe applies with children who are suffering from diseases and other debiliations and their families. I don't believe for a second that God intended that for them, but I also don't read in Scripture that God promises to heal and save people from physical pain or death. I believe that God is with many of those families and children, Christian families in particular because they are His. What I do read in Scripture is what Jesus thinks of death: When He came to the tomb of His friend, Lazarus, He wept. He wept even though He knew that He was going to raise him from the dead shortly after. I believe that He saw and felt what death does to the deceased's loved ones.
So if He is so moved by it, why doesn't He stop it? The Bible says that when Jesus returns He'll finish this creation and create a new heavens and new Earth where there will be no more pain, no more evil, no more suffering, and no more death because there will be no more curse on creation (that was placed on creation as a consequence of our sins). So why the wait? Peter explains that God is tolerating the current order of things for the moment because He doesn't want any to perish, but all to come to repentance and be saved. There will come a day when Jesus will return and do away with this world and its pain and restore it to it's former glory.
Just as an aside, I think that it is a mistake to assume that infants and children are 'innocent' and should thus be immune to the consequences of sin (namely death). All human beings (babies and children included) are sinners. You see a sinner isn't defined as such because they commit sins, rather, we sin because we are sinners ... it is in our nature to sin. Some verses which seem to indicate this include Psalm 51:5, John 3:16, and Romans 5:14. Infants, like the rest of us, are in a state of sin and need to be redeemed ... so no one is really 'innocent'. What is interesting is that Jesus warned against forbidding children to come to Him, which implies that children, like adults, need to come to Christ. That said, I believe that God is a compassionate God and that He is just in the way that He acts and as such He cannot hold infants and children accountable and so I believe that He will show them grace and allow them into heaven.
That's my two cents anyways...
All of that said, I don't think that we, as limited individuals, can understand the plans of God who sees the big picture (consider also that our lives, difficulties, and sufferings on this planet is so small when compared to eternity). We can only speculate as to how God acts and why He does the things that He does and why He allows the things that He allows. Ultimately we just have to trust that He is good and just and that He knows what He is doing and that one day it'll all become clear.