What your husband is going through is tough, but it's not unusual. Doubt is something that hits everybody--believers and unbelievers alike. The question about slavery, incest, and polygamy is a good one, because it goes to the heart of what God "allows" as compared to what He causes. To say God allows slavery, incest, and polygamy--not to mention murder and all manner of evil and suffering--is quite a different thing than saying God causes these things.
Now, from your brief description, it doesn't appear that your husband thinks God causes these things, but he is bothered that God allows them. In his mind, that may be just as bad. To put it another way, your husband is probably bothered--as are many people--by the fact that a loving God allows evil and suffering. To some people this is enough to discredit God's very existence.
So what do we do with evil and suffering? How do we reconcile bad things with a good God? The first thing we need to do is ask a very simple question: who's responsible for evil and suffering? Is it God, or are humans to blame? C.S. Lewis once said that four-fifths of the evil in our world is caused or done by humans. The other one-fifth of the evil is caused by nature (floods, tsunamis, earthquakes, etc.).
Still, that may not help your husband with his doubt. Even if you put the blame for evil squarely on humans and nature, why doesn't God prevent it? Couldn't God make people do the right thing? Couldn't He prevent nature from being destructive?
This goes to the heart of who God is and how He made us. The Bible tells us that God is love, and He made humanity in His image and then loved us so much that even when we were in rebellion against Him, He sent His only Son to give us the opportunity to be in a relationship with Him. God would have every right to destroy humanity, but He's letting us live. When God does finally deal with sin and suffering and evil, he will have to deal with sinners, once and for all. In the meantime, until Jesus returns to make all things new and to defeat death and sin forever, God is patiently giving people a chance to turn to Him.