For Question $3$ about the recurrence relations, using my code from
MMA question 285008
for $a_n := T_{7A}(n)$ I used
findseqrecur[4, 4, Array[t7A, 33, 1], 1, "a", k, -1] to get
$$ 0 = 14(n+1)(n+2)(2n+3) a_n \\ -3(n+2)(19n^2+76n+80) a_{n+1} \\ + 5(2n+5)(3n^2+15n+19) a_{n+2} \\ - (n+3)^3 a_{n+3}. $$
For $b_n := T_{7B}(n)$ there are several recurrences.
For degree $4$ polynomials I used
findseqrecur[6, 5, Array[t7B, 34, 1], 1, "b", k, -4] to get
$$ 0 = -7^5(k-17)(k-2)^3 b_{k-3}\quad \\ -7^3(19k^4-678k+2218k^2-2640k+1113)b_{k-2}\;\; \\ -7(85k^4-8707k^3+9978k^2-7072k+2090)b_{k-1} \\ +(85k^4+8707k^3+9978k^2+7072k+2090)b_{k}\, \\ +(19k^4+678k+2218k^2+2640k+1113)b_{k+1} \\ \quad +(k+17)(k+2)^3 b_{k+2}. $$
For degree $3$ polynomials I used
findseqrecur[8, 4, Array[t7B, 38, 1], 1, "b", k, -5] to get
$$ 0 = 7^7(k-3)^3b_{k-4}\; \\ + 7^5(47k^3-300k^2+646k-470)b_{k-3} \;\\ + 2\cdot 7^3(480k^3-1830k^2+2483k-1206)b_{k-2}\quad \\ + 7^2 (1578k^3-2001k^2+1513k-433)b_{k-1} \\ + 7 (1578k^3+2001k^2+1513k+433)b_{k}\;\; \\ \;\; + 2 (480k^3+1830k^2+2483k+1206)b_{k+1} \\ \; + (47k^3+300k^2+646k+470)b_{k+2} \\ \; + (k+3)^3 b_{k+3}. $$
Notice the symmetry of these two recursions.