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Coefficients are linear combinations of derivatives, derivatives are linear combinations of differences, differences are linear combinations of function values - hence point-wise convergence does imply coefficient-wise convergence.
"An infinite descending chain $T_{\alpha}$ would give an infinite ascending chain $I(T_{\alpha})$..." This is true only if $T_1\ne T_2$ implies that the closure of $T_1$ is different from the closure of $T_2$.If $T_1$ and $T_2$ have the same closure, then $I(T_1)$ and $I(T_2)$ are the same.
A sequence of polynomials converges point-wise to a polynomial if and only if it converges coefficient-wise to it. A linear space of polynomials is translation invariant if and only if it is invariant under partial differentiation.
