Sorry, my answer below is only partial, but I thought that it may still be somewhat interesting.
As far as I know, this inequality does not have a distinguished name. It is ultimately a consequence of the duality between $\|\cdot\|_\infty$ and $\|\cdot\|_1$, and the submultiplicativity of $\|\cdot\|_1$.
This is certainly known to you, but I wanted to remark that $$\|T\|_2^2 = \|T^*T\|_2 = \rho(T^*T) \le \|T^*T\|_1 \le \|T^*\|_1 \|T\|_1 = \|T\|_\infty \|T \|_1.$$ Thus, this inequality is nothing but an expression of:
- The C*-identity of the spectral norm
- Dominance of the spectral radius $\rho(\cdot)$ by the $\|\cdot\|_1$-norm.
- Submultiplicativity of the operator $\|\cdot\|_1$-norm.
As to the second inequality in the OP, we might need some further restrictions to make it more interesting. For instance, if $T$ is symmetric, then it reduces to $\|T\|_2 \le \|T\|_1$, whereby the ratio becomes $\|T\|_1/\|T\|_2$. This ratio has been studied for vectors (diagonal $T$) in the world of "sparse coding / compressed sensing, etc." --- see e.g., the paper here, and references therein.