Application and licencing fees apply to all entities, even those belonging to the crown. As for the naming of your station, though not something we would usually do, i personally would recommend the name; Dorset Station, after a historical earth Bretonian county which Poole was a part of
As for your pre-application for weapons platforms. As you suggested DSE has a long and positive relationship with the Kingdom and i see no reason why your application for 4 platforms would be refused, assuming you stick to the agreed distance from the current installations (trade lanes/stations/etc)