I am not a river, but I carry the same, though smaller in size, I still have my fame. What am I?

Beneath the ground I drink the rain, In a rabbit’s paw, I bring him fame.

Born as stone beneath the tide, I raise vast kingdoms side by side. Though unto thine eye I seem but rock, Countless little lives compose my stock. Handle me ill, and I wither away, Fading from the sea to mournful gray.

I am a bridge built without stone, I appear when the weeping sun is shown. I have seven colors but no voice, Finding my gold is a lucky choice.

I trap a rainbow deep in stone, October calls me as its own. Born of earth yet lit like flame, no two behold me quite the same.