You can own the Earth and still
This "the Earth" refers to the planet Earth (a particular entity), or in other words, this sentence means "You may be the owner of the world, but..."
All you'll own is Earth until
the article "the" is not used here. Although I feel the use of "the" here is optional and it would not make too much of a difference to the over all meaning even if it were included, this particular line is trying to decrease the value of "the Earth" that you were said to own in the previous line. The line is trying to imply as if the thing you own is just one of many, something very common and not too valuable. In other words, this sentence means "...you don't own anything of much value..."
You can paint with all the colors of the wind
I don't really know how to interpret this line for you but I think this would be one of those things that can be interpreted in many different ways, limited only by the interpreters imagination.
To rewrite the lyrics in slightly different words, "You may be the owner of the world, but you don't own anything of much value, until you have understood how to paint with all the colors of the wind".
"Once you can paint with all the colors of the wind, you'd stop owning
the earth"
This is the opposite (sort of) of what the original lyrics mean, and I suspect the confusion might be due to the somewhat poetic nature of the language used. In fact,
"you won't own the earth until you can paint with all the colors of
the wind"
is exactly what it is trying to say. The "not...until" relationship that you mention, can be attributed to the use of and still in the first line.