Do 1s & 2s orbitals overlap of different atoms while forming sigma bond? I think they should overlap as there will not be enough energy difference between them.
Orbital combinations involving inner shell electrons can indeed interact with outer-shell counterparts having the same symmetry. But, the inner shell orbitals are so deeply buried inside their respective atoms that the amount of overlap, and therefore the effect on bond/orbital energies, is very small. We find that in general, molecules are well modeled, without unneeded complexity, by including only outer shell orbitals. Moreover, any overlap involving inner shells, or even with outer shells, usually does not involve just $s$ orbitals. There can be $p$ orbital combinations and even $d$ orbital combinations having the same symmetry as the $s$ orbital ones, so the higher angular momentum orbitals also mix in. Only hydrogen and helium atoms with just $1s$ electrons can avoid that.
It isn't just our description of chemical bonding that exploits this property of inner shell electrons. We sometimes identify elements in a sample by hitting the samples with X rays that knock out inner shell electrons, and measuring the spectrum of absorbed energies. The energy required to knock out inner shell electrons from an atom is a characteristic of that element, and lack of strong overlap with other atoms makes that energy essentially invariant.