  Graph products of groups were introduced by Green in her thesis. They have an
operator algebraic counterpart introduced and explored by Fima and the
first-named author. In this paper we prove Khintchine type inequalities for
general C$^{\ast}$-algebraic graph products which generalize results by Ricard
and Xu on free products of C$^{\ast}$-algebras. We apply these inequalities in
the context of (right-angled) Hecke C$^{\ast}$-algebras, which are deformations
of the group algebra of Coxeter groups. For these we deduce a Haagerup
inequality. We further use this to study the simplicity and trace uniqueness of
(right-angled) Hecke C$^{\ast}$-algebras. Lastly we characterize exactness and
nuclearity of general Hecke C$^{\ast}$-algebras.
