Why is the graphite moderator in a nuclear reactor radioactive?
Graphite in reactors gets radioactive mainly by forming beta decaying $\require{mhchem}\ce{ ^{14}_6C}$, mostly from naturally occuring stable $\ce{^{13}_6C}$ (1.1% abundance) :
$$\ce{^{13}_6C + ^{1}_{0}n -> ^{14}_6C}$$
with half-life $5730 \pm 40$ years. It is naturally present in traces in all carbon containing matter with recent carbon interchange with atmosphere.
The beta decay has the equation:
$$\ce{^{14}_6C -> ^{14}_7N + ^{0}_{-1}e + \overset{-}\nu_{\mathrm{e}}}$$
It is the same isotope used for radiocarbon dating, as it is continuously created in atmosphere by cosmic radiation.
Heavier carbon isotopes are very unstable and quicky undergo beta decay to nitrogen ( or even oxygen ).
Secondary source of radioactivity is contamination by fission products and radioactive products of neutron irradiation of stable isotopes.
For the former, there is wide range of formed radioactive isotopes, with 2 distribution peaks with relative masses about 2/5 ( like $\ce{^{90}Sr}$ ) and 3/5 ( like $\ce{^{131}I ^{137}Cs}$ ) of the mass of $\ce{^{235}U}$.
If you used graphite control rods, you would cause the 2nd Czernobyl.
RBMK uses graphite tipped control rods, not graphite control rods.
These tips cause problem in short reversal of control rod effects.
Graphite is not used as control rods, but as a neutron moderator, similarly as light or heavy water.
Moderators serve for slowing down neutrons to thermal speed(thermal neutrons) to improve the cross-section of the fission reaction, so the sustained fission reaction is possible even in lightly enriched or even natural uranium(heavy water nuclear reactors). Best moderators have low neutron absorption.
Moderation principle is on purely mechanical bases, as fast neutrons collides with slow particles (protons, deuterons, carbon kernels) and collisions redistribute momentum.
From mechanical point of view, protons would be best moderators, but unfortunately they considerably fuse with neutrons, so enriched uranium is needed ( in the opposite to deuterons )
Control rods are the opposite. They serve for regulation of "neutron economy" by absorption of excessive neutrons.
There are usually used 3 sets of rods,
Regulation set for BAU regulation
Adjusting sets, progressively pulled out, as neutron economy gets worse in the time
Emergency set, that stops the fission when triggered to be pushed in.
The rods contain usually cadmium or preferably boron, with high cross-section for neutron absorption.
Boric acid is also used in cooling water in secondary/tertiary circuits, where water serves as heat transfer medium only, not as moderator.