1/4 inch plywood will easily wrap unless you build a frame of some kind to support it.  This is not suitable for outdoor use, it does not hold up well to extreme weather. It will get warped and the wood will begin to split.

1/2 inch plywood might get few problems specially during hot weather. Some of the problems you will see is the laminate will start to break away and warp.  The thinner layers are more likely to do this.  If the item is well sealed and painted (especially the edges) it should last with only minor touch-up.

Best is 3/4 inch thickness for exterior use. This plywood usually will not warp.

The ultimate goal is to complete the projects in the most cost effective manner. And, to have the finished project last for many years to come. So best to use 3/4 inch plywood.