Matrix Spectral Radius and Induced Matrix Norms

Let $A$ be a matrix, and $\rho(A)$ be its spectral radius, $\|A\|_p$ be an norm induced from vector $p$-norm.

(1) When $\rho(A)=\|A\|_2$ or $\rho(A)=\|A\|_1$, does $\|A\|_1=\|A\|_2$?

(2) If the answer to question (1) is in the negative, does making $A$ normal or Hermitian imply $\|A\|_1=\|A\|_2$?

(3) What is an example of positive matrix $A$ where $\rho(A)<\|A\|_2$ or $\rho(A)<\|A\|_1$

(1): No. Take $$ A = \pmatrix{-1&1\\1&1} $$ We find $\|A\|_2 = \rho(A) = \sqrt 2$, but $\|A\|_1 = 2$.

Conversely, take $$ B = \pmatrix{2&1\\0&1} $$ We find $\|A\|_1 = \rho(A) = 2$, but $\|A\|_2 = \sqrt{3 + \sqrt5}$.

(2): No, see the above example.

(3): In addition to the above example, take $$ A = \pmatrix{1&1\\\epsilon&1} $$ For some small $\epsilon>0$. We have $\|A\|_2 \approx \sqrt{\frac{3 + \sqrt5}{2}}, \|A\|_1 = 2, \rho(A) \approx 1$.

Or, for a more particular example, take $$ A = \pmatrix{2&2\\1&2} $$ We have $\rho(A) = 2 + \sqrt 2$, $\|A\|_1 = 4$, and $\|A\|_2 = \sqrt{\frac{13+3 \sqrt{17}}{2}}$