For the first question, $n^{1/2}$ is the right order for $p=1,2,\infty$. The case $p=\infty$ is arguably the easiest, because $B(\ell_\infty^n)=\ell_\infty^n(\ell_1^n)$ isometrically and the Banach-Mazur distance between $\ell_\infty^n$ and $\ell_1^n$ is of order $n^{1/2}$. That gives the upper bound, and the lower bound is also true because $\gamma_\infty(\ell_1^n)$ (the factorization constant of the identity on $\ell_1^n$ through an $\ell_\infty$-space) is of order $n^{1/2}$.

$$
$$
Also $B(\ell_1^n)=\ell_\infty^n(\ell_1^n)$ isometrically, and you get the lower bound from the fact that $\gamma_1(\ell_\infty^n)$ is of order $n^{1/2}$. For the upper bound assume that $n$ is a power of two (I think standard arguments from this case give the general case but did not try to think it through). In $L_1^N(\ell_1^n)$ consider the basis $w_k\otimes e_j$, $1\le j,k \le n$, where $(w_k)$ is the Walsh basis for $L_1^n$. Take the basis to basis mapping from $L_1^N(\ell_1^n)$ onto $\ell_\infty^n(\ell_1^n)$ that maps $w_k\otimes e_j$, $1\le j \le n$, onto the $k$-th copy of $\ell_1^n$. This mapping has norm at most $n^{1/2}$ because the Walsh basis has an upper $\ell_2$ estimate, and the inverse has norm one.
$$
$$
For $B(\ell_2^n)$, Yemon's comment gives a lower estimate of $n^{1/2}$ (in fact, for all $1\le p \le 2$ because $\gamma_p(\ell_\infty^n)$ is of order $n^{1/2}$ in this range). The upper estimate is just the fact that the norm of an operator in $\ell_2^n$ is at most $n^{1/2}$ times the Hilbert-Schmidt norm of the operator.
$$
$$
For all values of $p$, if you write a vector in $\mathbb{R}^{n^2}$ as an $n$ by $n$ matrix $A$, then the $\ell_p^{n^2}$ norm of $A$ is the $p'$-summing norm of $A$ considered as an operator from $\ell_p^n$ into $\ell_{p'}^{n}$. That is what is used above in the $p=2$ case, but I don't see that this helps for other values of $p$.
For $2<p<\infty$ you get a lower bound from the fact that $B(\ell_p^n)$ contains $\ell_\infty^n$ and $\ell_{p'}^n$ isometrically, but for $p=4$ that gives a lower bound of only $n^{1/4}$.

`$<\infty$`

rather than`$=\infty$`

in your second question. As for the first question, probably someone computed the GL constant for`$B(\ell_p^n)$`

. That might give the order of magnitude for the B-M distance you are seeking. I would look in Tomczak's book but my copy is loaned out. $\endgroup$`$n^{1/p}$`

for`$p\ge 2$`

and`$n^{1/2}$`

for`$1\le p\le 2$`

,Yemon (using cotype constants on $n$ vectors). $\endgroup$