I'm refering to the question in the title. Elements such that $\chi(g)\neq 0$ for all $\chi\in \operatorname{Irr}G$ are called nonvanishing elements in the literature. They were studied by Isaacs, Navarro and Wolf (Group elements were no irreducible character vanishes, J. Algebra 222 (1999), 413-423, MR1733678, Doi) and others (search for papers that cite the paper just mentioned). Isaacs, Navarro and Wolf prove (among other things) that in a nilpotent group, all nonvanishing elements are central. On the other hand, Miyamoto (J. Algebra 364 (2012), 88-89, MR2927049, doi) showed that any nontrivial solvable group has nonvanishing elements different from $1$. Thus at least every solvable groups with trivial center is a group as in the question title. A paper of Dolfi et. al. (J. Algebra 323 (2010), 540-545, MR2564856, doi) contains a number of nonsolvable examples.