Are all Karstarks named after Starks

The Arryn Valley, almost completely framed by the Moon Mountains, is a region in the east of Westeros. It has always been under the rule of House Arryn. The headquarters of the House of Arryn is the Eyrie.
A trip to the valley of Arryn is possible by sea or, more often used but also far more dangerous, by land. Anyone planning the route over the passes of the Moon Mountains should think about the tribes of the Moon Mountains beforehand. Their life, their culture is simple and original. Their approach is all the more determined and aggressive when it comes to defending their clans and their territory. For some time now, clan warriors of the Moon Mountains have become increasingly dangerous. They raid travelers and trade routes. In addition, these clans live permanently in feud with each other and with the houses of the valley.

The well-known Westeros clans in the Arryn Valley area not only have significant names, they all have their own very strange and unsettling rituals. For example, the clan warriors cut the Black Ears, under the leadership of Chella, daughter of Cheyk, remove an ear from each of their fallen enemies and wear them as trophies.
The Burned Men are warriors who are only considered real warriors after the ritual of self-mutilation by fire. Led by Timett, Timett's son, they spread fear and terror in the valley.
Shagga, son of Dolf, is the clan leader of the Stone crows. This clan fights as simple as it is bloody: Frontal attacks that resemble wild frenzy also cost the lives of the best-armed opponent.

There are many other clans in the Arryn Valley and Moon Mountains. The Milk snakes, the Moon Brothers, the Painted dogs or the Sons of the Mist. They all form the clans of the Moon Mountains who, like guerrillas, fight the passes and travel routes and enemy armies and thus represent an everlasting danger.

From the Clan Warriors of the Mountains: Units & Commanders