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.