How to tell if 2×4 is untreated and safe for indoor project

You won't want to use pressure treated wood in any application where people will be touching the wood often, especially children. Cedar is a good wood for outdoor applications, and is often used in playhouses and swing sets. If you've got really deep pockets, Teak is a very good choice for outdoor applications. Teak is often used in higher end patio furniture, because is is very low maintenance and stands up well to weather, due to its high oil content much like cedar.

You could probably get away with any type of wood (even pine), but you'd have to seal it yearly and replace pieces if they did start to rot.

A month or two. Hard to say I've seen them almost dripping but I've also gotten them nice and dried out because they were probably sitting around on site or at the lumber year for a bit. They key would be good airflow don't just keep them in a pile.

I too would recommend putting the fence up then painting it later