For Question $3$ about the recurrence relations, using Mathematica,
for $a_n := T_{7A}(n)$ I found:

$$ 0 = 14(n+1)(n+2)(2n+3) a_n \\
 -3(n+2)(19n^2+76n+80) a_{n+1} \\
 + 5(2n+5)(3n^2+15n+19) a_{n+2} \\
 - (n+3)^3 a_{n+3}. $$

For $b_n := T_{7B}(n)$ I found:

$$ 0 = -7^5(n-14)(n+1)^3 b_n \\
-7^3(19n^4 -1450n^3 -2858n^2 -5586n -3612)b_{n+1} \\
-7(85n^4 -7687n^3 -63795n^2 -173113n -157528)b_{n+2} \\
+(85n^4 +9727n^3 +92931n^2 +311209n +355082)b_{n+3} \\
+(19n^4 +906n^3 +9346n^2 +36306n +48840)b_{n+4} \\
+(n+20)(n+5)^3 b_{n+5}. $$