We apply density functional theory to study the effective interaction between dopant atoms (B, N, Si, P) and functional groups (H, F, Cl, OH) on the surface of carbon fullerenes. Both dopant atoms and functional groups strongly interact through the carbon cage even in diametrically opposite positions. Interaction energies distribute in a wide range from 0.1 to 2 eV and non-monotonically depend on fullerene size and distance between dopants or functional groups. Such interaction cannot be described as a simple Coulomb repulsion or sum of dopants binding energies and cage strain energy. We identify some general trends in relative positions of dopants or functional groups in low-energy isomers. Para position of two functional groups is the most feasible for C60 and larger cages. For lower fullerenes, ortho or other spaced positions may be more preferable. The interaction of foreign atoms embedded into the carbon cages is more complicated. The best relative positions intricately depend on the cage size and chemical nature of dopants. As a rule, ortho and para locations are feasible for C60 and larger cages. However, some exceptions are observed. The effect of thermal vibrations on the considered interactions in doped or functionalized fullerenes is negligible in the temperature range from 300 to 1000 K.