Are the outer electrons of any atom responsible for doing the bonding between atoms?
Yes the outer electrons called valance electrons and only the outer electrons are involved in bonding between atoms and molecules.
The sharing of electron density between atoms creates the chemical bonds between atoms that form molecules and ionic crystal structures. It is only the outmost or valance electrons that are able to be shared to form bonds. iThe outermost electrons can be only the s electrons. or a combination of s and p electrons. or a combination of s and d electrons from a shell one lower than the s electrons.
The sharing of electron charge forms intermolecular forces such as Hydrogen bonding, dipole dipole and Van de Waals attractions. These attractive forces cause molecules to join together forming solids and liquids. Again it is only the outermost electrons that create these intermolecular forces.