This inverse limit group scheme is usually called $\mathbf{Z}_p(1)$. The symbol $\mu_{p^\infty}$ is usually reserved for the direct limit (as sheaves) of $\mu_p \subset \mu_{p^2} \subset ...$ along the usual inclusion maps.

For the second question: note that you have $T_p(K) =~ \widehat{K}[-1]$, where $\widehat{K}$ denotes the derived $p$-adic completion of $K$. It is well-known (and easy from derived Nakayama) that $\widehat{K} =~ 0$ if and only if $K \otimes_{\mathbb{Z}}^L \mathbb{Z}/p \simeq 0$. Using the standard resolution, it follows that $T_p(K)$ has $0$ cohomology if and only each each $H^i(K)$ is uniquely $p$-divisible.