A redated post.
As for how it is possible to believe that humans ought to alleviate suffering but God may be justified in permitting it, it is to be remembered that humans have a far narrower purview of considerations than God does. We do the best we can with no knowledge, for example, of how our actions will affect the free choices of people living 50 years from now, but God knows how those choices will be affected and may choose to do something that seems bad now in order to improve the situation with respect to the situation 50 years from now and 100 years from now.
If we eliminate certain diseases, we cannot be absolutely sure that they will not make the world a worse place in the long run. But to the best of our knowledge, they will cause an improvement of human life. Would you cure smallpox if you knew that if you didn't cure it, Hitler's mother would die of smallpox and the world would be spared the Holocaust? You don't have those things to think about, because you are a human with limited knowledge. God's knowledge is not limited (unless you're an open theist, in which case God's knowledge is still a whole lot less limited than yours is!)