What is the reference length in calculating the friction coefficient?
Since the site computes atmospheric conditions, only the friction coefficient $c_f$ needs a reference length. This is the length of the object causing friction, measured along the direction of airflow. In case of a wing, it is its local chord, in case of a fuselage, it's the fuselage's length. 

Each part of an airplane and each section of the wing will have a slightly different $c_f$, depending on local chord length. The highest contribution to friction drag is caused at the leading edge, when the boundary layer is thin and the speed gradient at the surface is steep. The farther downstream you look, the less each additional unit of length will add to friction drag, so the average over the full length (that is what $c_f$ stands for) will become smaller for longer bodies.

To compute total drag, you need to add the contribution of each part, multiplied with its surface area relative to the airplane's total area. Yes, for a wing the upper and the lower wetted surface area will each contribute their own drag component. Since airflow is faster on the upper surface of a lift-creating wing, you would need a correction factor to account for the speed difference between upper and lower side, but if the difference is small, you can just assume the same speed everywhere. To be precise: $c_f$ is the friction coefficient of a flat plate parallel to the airstream.