Yes, this is an observation of H. Blichfeldt ( I think J-P. Serre also attributes it to Minkowski), which has been rediscovered many times over the years. Letting $1$ denote the trivial character, note that $\theta = \prod_{i = 1}^{\ell} ( \chi - \chi(c_{i})1) $ is an algebraic integer combination of characters of $G$, and also vanishes everywhere on $G$ except the identity, where its value is $\prod_{i = 1}^{\ell} ( n - \chi(c_{i})) .$ Hence $\theta$ is an algebraic integer multiple of the regular character of $G$, and $\langle \theta,1 \rangle = \frac{1}{|G|}\prod_{i = 1}^{\ell} ( n - \chi(c_{i})) $ is an algebraic integer ( which is easily seen to be rational by Galois theory, hence is a rational integer).

Yes, this is an observation of H. Blichfeldt ( I think J-P. Serre also attributes it to Minkowski), which has been rediscovered many times over the years. Letting $1$ denote the trivial character, note that $\theta = \prod_{i = 1}^{\ell} ( \chi - \chi(c_{i})1) $ is an algebraic integer combination of characters of $G$, and also vanishes everywhere on $G$ except the identity, where its value is $\prod_{i = 1}^{\ell} ( n - \chi(c_{i})) .$ Hence $\theta$ is an algebraic integer multiple of the regular character of $G$, and $\langle \theta,1 \rangle = \frac{1}{|G|}\prod_{i = 1}^{\ell} ( n - \chi(c_{i})) $ is an algebraic integer ( which is easily seen to be rational by Galois theory, hence is a rational integer).

(Edit: In fact, it is only necessary to take the distinct values of $\chi(c_{i})$ without repetition in the product which is what Blichfeldt did).