Is it normal for a presta valve to unscrew into two pieces?
The correct answer is yes and no. There are in fact two types of Presta valves: One where the core can be removed. Those are often used with tubular tyres so that the inner tube can be filled with a sealant (latex) fluid. Some higher quality tubes also have removable core valves for the same purpose. Normal and cheaper tubes have Presta valves where the core cannot be removed.

There is a special tool to be used for handling the core although like @Michael said the (careful) use of pliers also works.

Another reason for a removable core is the possibility to use valve extensions with high rims.