Damped Oscillation and Period

Yes, damping forces like friction and drag force affects period in way : $$ T = {2pi } left( sqrt{frac{k}{m} - left(dfrac{b}{2m}right)^{2}} ~right)^{-1} $$

Where $b$ is damping coefficient.

The free vibration frequency is affected by damping.

If you assume the damping force is proportional to velocity, the math is well known. See http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html for example, particularly the difference between the damped frequency $omega_1$ and the undamped $omega_0$ in the last equation.

The answer in the link you posted is hand-waving, not rigorous math, but it gives the right physical idea of why the frequency changes.

You can find the amount of damping by measuring how much the amplitude of oscillation decreases from one cycle to the next.

For a pendulum swinging in air, the damping force is not proportional to the velocity (it is more likely to be proportional to the square of the velocity) but for small amounts of damping, you can still fit a curve of the form $x = A e^{-gamma t}cos omega_1 t$ to the measured displacements and then find the undamped frequency using $omega_0^2 = omega_1^2 + gamma^2$ as in the hyperphysics link.

For a pendulum, the change in frequency is likely to be small, but it is still worth checking if the correction to your experimental data is significant or not.