[Electron]

The preamp power supply is to be installed at a place where the weather can not get to it.
It is not required to be indoors , but out of the weather.
And also the power supply plugs into a AC outlet and mostly AC outlets are mostly indoors.

The actual preamp part is made to be in the weather or out of the weather.
The actual preamp is usually installed close to the antenna so as to amplify the signal before any signal loss through long runs of coax.

Most of the boxes that have coax in them are on the outside wall of the house usually close to electric service and breaker box , and the phone box.

As an example.
If the location is a outside garage wall then what I often do is simply have the coax and connections on the opposite of the outside wall , the inside wall of the garage.

Yes the splitter can go in the box where the coaxes are at.