Why is it that If I leave a fondant cake out it is fine, but when it is half eaten and I pack it into a box or closed dish, so that it doesn't go stale, it seems to melt and go all sticky!

I'm pretty sure if you have it in a closed dish, the cake is going to "sweat" which will cause the fondant to become sticky. There was a thread a couple days ago about "the science of baking" and i believe they were talking about this. You should check it out, it may help