Who was it ruled by?

In prehistory, we have evidence that there were humans living in modern day Normandy. After that, the Celts and Gauls had control of Normandy, before being conquered in the Gallic Wars. At this point, Normandy is now romanized.

In the late third century, we have evidence of barbarian tribes raiding Normandy. Coastal settlements of Normandy in particular were in danger of Saxon raids.

Near 486, Clovis I, ruler of the Franks, conquered Normandy. Around 790, the Vikings started their raids on Normandy and at that point you apparently know.

What was their House?

The Celtic Tribes who ruled were numerous, the list is easily accessible. Normandy was incorporated into Rome by Augustus, of the Julio-Claudian Dynasty. Clovis I, of the Franks, belonged to the Merivingian Dynasty.

What was it called?

The Romans considered it part of Brittany until Diocletian detached it, while Brittany itself was a region of the province Gallia Lugdunensis. The Franks considered it part of Neustria, the name for their Western territory.

What happened to them?

Well the Gauls were conquered and by the time Clovis I took it from Rome, it was almost gone anyways.