**1)** No, in general, convergence does not hold because of the presence of spurious poles. A typical result, see [1], is

**Theorem.** Let $(n_{\nu})_{1}^{\infty}$ be a sequence of positive integers satisfying $n_{\nu}>2 n_{\nu-1}$
for all $\nu$. Then there exists an entire function
$$
f(z)=\sum_{i=0}^{\infty} a_{i} z^{i}
$$
with $a_{i}$ tending to 0 arbitrarily fast, so that the $\left(n_{v}, n_{v}\right)$ Padé approximant to $f$ has a pole at any prescribed point $b_{\nu} \neq 0$ of the complex plane, for
$\nu=1,2, \ldots$

**2)** The exponential function is an entire function for which local uniform convergence of the diagonal Padé approximants $R_{n,n}$ is known (this is due to Padé). The rate of convergence at a point $z=x+iy\in\mathbb{C}$ is given by
\begin{equation*}
|e^z-R_{n,n}(z)|=\frac{(n!)^{2}}{(2n)!(2n+1)!}|z|^{2n+1}
e^{x}(1+o(1)).
\end{equation*}

[1] H. Wallin, The convergence of Padé approximants and the size of the power series coefficients.
Applicable Anal. 4 (1974), no. 3, 235-251.