Does house Arryn have a look like the lannisters, starks, baratheons, and tullys do?

Yes. We know this from when Sansa meets Harry the Heir:

Ser Harrold Hardyng looked every inch a lord-in-waiting; clean-limbed and handsome, straight as a lance, hard with muscle. Men old enough to have known Jon Arryn in his youth said Ser Harrold had his look, she knew. He had a mop of sandy blond hair, pale blue eyes, an aquiline nose. Joffrey was comely too, though, she reminded herself. A comely monster, that’s what he was. Little Lord Tyrion was kinder, twisted though he was.

Blond hair and blue eyes attest to the Arryns’ Andal heritage, but the aquiline nose I think is the more specific detail, evoking the curved beak of the falcon which the Arryns took for their sigil. 

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