Because the names are figurative, not literal, at least given the evidence that makes the most sense,
water refers to liquids such as water or liquid protodermis, while ice refers to freezing, and manipulating ice-like structures, such as the crystalline protodermis of ko-metru.
Stone is surface rocks, boulders and sand, earth is soil and the sediment.
I'm fairly certain toa of fire can only absorb flame and not pure heat,
Just as toa of ice can't absorb pure cold.
Well, a toa of ice can freeze water, a toa of water has no kinesis over ice.
It is and that was addended in g2, making sand not be controlled by stone in g1 was a mistake.
Yes, in g1 it was referred to as 'the green'.
As far as I can tell, Greg wanted more elements and those seemed cool. I'm not sure if anyone ever asked Farshtey his reasoning.
If I got any lore wrong I'm sure some archivist on the boards will correct me.