For question 3.
Using Maple's gfun library, I get this $4$-term recurrence for $t_{7A}$: \begin{align} 0 &=14\, \left( 2\,n+3 \right) \left( n+2 \right) \left( n+1 \right) u \left( n \right) \\ & -3\, \left( n+2 \right) \left( 19\,{n}^{2}+76\,n+80 \right) u \left( n+1 \right) \\ & +5\, \left( 2\,n+5 \right) \left( 3\,{n }^{2}+15\,n+19 \right) u \left( n+2 \right) \\ & - \left( n+3 \right) ^{3}u \left( n+3 \right) . \end{align}
I will work on $t_{7B}$, but for some reason that triple summation seems hard for Maple.