A coupled cluster (CC) investigation is presented for the (generalized) Sternheimer shieldings and the electric field gradient (EFG) polarizabilities which describe the effect of external electric fields and field gradients on the electric field gradient at the nuclei. Calculations are performed for the linear molecules N-2, CO, HF, C2H2, HCl, HCN, and HNC. Correlation effects are monitored by employing a hierarchy of CC models consisting of CCS, CC2, CCSD, and CC3. The effect of tight basis functions and core correlation is investigated by carrying out CCSD calculations with core-valence basis sets. Accurate theoretical estimates for EFGs, Sternheimer shieldings, and EFG polarizabilities are given and the effects of vibrational corrections are discussed. Our final estimates for the considered EFG properties can be used, for example, in simulations of electric field effects on the EFG at the nuclei in interacting molecules.