The centroid of the zeros of the kth partial sum of exp(z) is -1?

A) About your first question: you can attempt an induction proof but it is simpler to make a direct use of one of the Vieta's formulas: the sum of the roots $r_1,\cdots r_n$ of $\sum_{k=0}^n a_kz^k=0$ is equal to

$$\sum_{k=1}^n r_k=-\frac{a_{n-1}}{a_n}\tag{1}$$

Knowing that here $a_k=\dfrac{1}{k!}$, (1) becomes, divided by $n$:

$$\dfrac{1}{n}\left(\sum_{k=1}^n r_k\right)=-\dfrac{1}{n}\dfrac{1}{(n-1)!}n! =-1$$

B) The essential property

$$P'_k(z)=P_{k-1}(z)$$

has an immediate consequence: the "russian dolls" inclusion of the successive convex hulls due to the classical Gauss-Lucas theorem. Here is a graphical representation of the roots and their convex hulls; the result is rather amazing (see below the Matlab program that has generated this figure)

Successive convex hulls of the roots of the $P_k$s, from $k=2$ to $k=25$ (the largest one). "Central" point $(-1,0)$ is materialized by a star.

This figure is of course symmetrical with respect to $x$ axis, due to the fact that roots of polynomials with real coefficients are either real or are pairs of conjugates.

Other properties can as well be observed, deserving further analysis, dealing in particular with your second question about the module of the roots with largest module (situated on a curve very close to a parabola).

A deep analysis of these roots is conducted here; observe the final semigraphical representation, surely the best one could obtain with printers 50 years ago, a testimony of the state of the art at that time, even with IBM labs top equipment...

See as well this StackExchange question and its answers, using Lambert $W$ function.

Matlab program:

clear all;close all,hold on;
plot(-1,0,'pk');
plot([-1,-1],[-1,1],'b');C=[1/2,1,1];% case k=2
n=25;
for k=3:n; % k=degree
    C=[C(1)/k,C];
    r=roots(C);R=real(r);I=imag(r);
    H=convhull(R,I);
    plot(R(H),I(H),'b');
    plot(R,I,'.r');
end