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}
It seems this is different from the Somos recurrence. So these should be examined more carefully.
Machine-readable:
2401*(n+3)*(n^2+6*n+35)*(n+1)^3*u(n)+(1274*n^6+17199*n^5+129311*n^4+551299*n^3+ 1264592*n^2+1459808*n+668850)*u(n+1)+(267*n^6+4005*n^5+32065*n^4+153775*n^3+ 421280*n^2+601400*n+346290)*u(n+2)+(26*n^6+429*n^5+3614*n^4+18779*n^3+57893*n^2 +94588*n+62265)*u(n+3)+(n+2)*(n^2+4*n+30)*(n+4)^3*u(n+4)