 $\sin\theta=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\frac{\theta^9}{9!}-\frac{\theta^{11}}{11!}+...$
The error due to approximation can be estimated as $\frac{\theta^3/3!}{\theta}=\frac{\theta^2}{6}$
For 5 degree, the error is about $\frac{(5\pi/180)^2}{6}=0.00127= 0.127%$ which is a very small error.
For 10 degree, the error is about 4*0.127%= 0.50%
For 20 degree, the error is about 16*0.127=2.0%

What I want to point out is:
1. 5 degree is not a magic number: user should provide the precission required in order to determined the maximum angle for good approximation.
2. Even 20 degree only produce 2% of error which is normally smaller than experiment error when student perform real  experiment.

The solution for RLC circuit is: $Q(t)=Q_0 e^{-\alpha t}e^{i\omega t}$
where $\alpha=\frac{R}{2L}, \omega=\sqrt{\frac{1}{LC}}$
Because $\alpha$ is independent of charge or current: i.e. damping rate is the same for the same R/L value.
You should be able to find out solution for pendulum(compare equations for RLC and pendumum set)