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For your question $2$ a quick search of 'eta07.gp' turned up $t_{12,12,40}$ and the other $5$ members of the $6$ cluster. They are: $$t_{12,12,40},\; t_{12,12,56},\; t_{12,24,72},\; t_{12,24,90},\; t_{12,24,120},\; t_{12,24,126}.$$ For the first, the identity is equivalent to $$ (\sqrt{1/4}u_1^3u_{12}/(u_3u_4^3))^2 + (\sqrt{3/4}u_2^4u_6^2/(u_1u_3u_4^4))^2 = 1 $$ and similarly for the other $5$ identities. Probably other similar clusters can be found.