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}

and this $5$-term recurrence for $t_{7B}$:
\begin{align}
 0 &= 7^4\, \left( n+3 \right)  \left( {n}^{2}+6\,n+35 \right)  \left( n+1
 \right) ^{3}u \left( n \right) 
\\ &+ \big( 1274\,{n}^{6}+17199\,{n}^{5}+
129311\,{n}^{4}+551299\,{n}^{3}
\\ &\qquad\qquad+1264592\,{n}^{2}+1459808\,n+668850
 \big) u \left( n+1 \right) 
\\ &+ \big( 267\,{n}^{6}+4005\,{n}^{5}+
32065\,{n}^{4}+153775\,{n}^{3}
\\ &\qquad\qquad+421280\,{n}^{2}+601400\,n+346290
 \big) u \left( n+2 \right) 
\\ &+ \left( 26\,{n}^{6}+429\,{n}^{5}+3614\,
{n}^{4}+18779\,{n}^{3}+57893\,{n}^{2}+94588\,n+62265 \right) u \left( 
n+3 \right) 
\\ &+ \left( n+2 \right)  \left( {n}^{2}+4\,n+30 \right) 
 \left( n+4 \right) ^{3}u \left( n+4 \right) 
\end{align}