Roku - fire
Kyoshi - earth
Kuruk - water
Yangchen - air
we dont get told anyone else after that
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Too many to count. In the second or third episode, you will see inside the air temple the statues of hundreds of thousands of avatars before Aang.
The arrow tattoos are for airbenders only, not all Avatars. Aang got them because his natural element is air, and they get the tattoo to show everyone he is a master airbender. Roku was a firebender originally, so of course he will not get tattoos.
Aang is Korra because they said the avatar spirit passes down.
Avatar of air? I think you mean Avatar who was born into the Air Nomads. And to answer your question, it hasn't been revealed, and it is unlikely to be revealed. But the two most recent of them are Avatars Aang and Yangcheng.
Between Aang, Roku, and Kyoshi, it seems that on average, each Avatar lives about 152 and a half years, so in a thousand years, there must've been about 6 or 7.