Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$

A somewhat more general setting, namely, finding the best constant $C_{p,q,r}$ in \begin{equation*} \|AB-BA\|_p \le C_{p,q,r}\|A\|_q\|B\|_r, \end{equation*} for Schatten $p$,$q$,$r$-norms, is studied in this paper.

EDIT (1st Mar'17). See also this recent paper (LAA, 521(15), May 2017, Pages 263–282) that provides sharp bounds on the constant $C_{p,q,r}$ above for real matrices.

Concerning your first question, as I noted in my comment above for any operator norm we have $\Vert [A,B]\Vert \leq 2\Vert A\Vert\Vert B\Vert$.

Conversely, let $A = \pmatrix{1 & 0 \\ 0 & -1}$, $B = \pmatrix{0 & 1\\1 & 0}$ so that $[A,B] = \pmatrix{0 & -2\\2 & 0} = 2\pmatrix{0 & 1\\-1 & 0}$.

Then for all $1\leq p \leq \infty$ $\Vert A\Vert_p = \Vert B\Vert_p = 1$, and $\Vert [A,B]\Vert_p = 2$