Regarding the bounds for the Picard number $\rho$, the lower bound $\rho \geq 1$ comes from the fact that there exist divisors which are not algebraically equivalent to 0 (alternatively, one can argue that $\operatorname{id}_A$ is a symmetric endomorphism of $A$). The upper bound $\rho \leq (\dim A)^2$ (which holds only in characteristic 0) is Exercise 2.6(5) in Birkenhake-Lange's book Complex abelian varieties.
It is possible to show further that if $A$ is an abelian variety over $\mathbf{C}$ of dimension $\geq 2$, then $\rho = (\dim A)^2$ if and only if $A$ is isogenous to a power of some CM elliptic curve (see Exercise 5.6(10) in the same book).