Is there a gauche interaction between C-3 and C-6 in cyclohexane?
Yes, you're exactly right. Not only is the axial methyl gauche with respect to the methylene unit that you point out, but there are also gauche interactions of the two methylenes that are drawn in the "bridging" positions of your Newman projections.

So, why is the chair conformation so stable?

For one, gauche interactions are really not that destabilizing. You're talking about maybe a few kJ mol-1. Realistically, the axial interactions between that methyl group and other axial hydrogens is going to be more energetically unfavorable. But this is easily solved by a ring flip, which maintains the chair conformation.

Whoever mentioned that chair cyclohexane "acts like a long carbon chain" was probably referring to the fact that the bond angles are close to 109.5 degrees, which means that it is far less strained than rings that are fewer than 5 carbons or greater than 7 carbons. However, that's true for boat cyclohexane as well, so it's not that relevant to the question at hand.

For reference, boat cyclohexane is about 25 kJ mol-1 higher in energy than chair cyclohexane, so the gauche interactions you're talking about are a very minor contribution the the energy of the molecule.