Actually the formula is (3 * #of atoms)-(# of bonds) This is what i found to be it in a few other videos as well and it works all the time but bear in mind there are exceptions in some cases. I don't understand why here they have used a different formula.

The degrees of freedom are not just limited to translational degrees of freedom, there are vibrational and rotational degrees of freedom too. However we tend to disregard the vibrational degrees of freedom in a gas as its insignificant

Vibrational DOF in diatomic molecule is absent,, hence just 5 DOF, same thing happens in case of all diatomic molecules

vibrational degrees of freedom at room temperature do not count because vibrational energy levels can be modified only at higher temperatures where collisions are sufficiently violent for vibrations to start. (see quantum mechanics) At low temperatures even rotational DOF can be neglected (bc. of slower motions do not result collisions violent enough to make molecules rotate) At room temperatures mainly translational and rotational DOF contribute to Cv.