Coincidence, purposeful definition, or something else in formulas for energy

Usually, linear equations are very common in physics. Something like $\text{Quantity}=\text{constant}\times\text{variable}$

So, you have $p=mv,L=I\omega,Q=CV$, etc.

Now, it just turns out that when you multiply these linear equations with a small increment of the variable, you get an expression for energy. Why this happens isn't so easy to figure out. The easiest explanation is that we usually define our variables such that force and similar quantities (e.g, net charge, etc) are linear.

Now, if you sum up a linear equation multiplied by a small change in the linear part, you get something like $\int cx dx$, which is a quadratinc term of the form $\frac12 cx dx$

Most of them (all of your examples except $E=c^2m$, which is really just $E=m$ anyway) arise from integrating a linear equation like $p=mv$ as $E=\int v\,dp$, and it is often just a convention that we choose the linear relation to have a constant of proportionality of 1, so the integral has a constant of 1/2 (for example, we could've instead chosen, like we do with areas of circles, to have $c=2\pi r$ and $A=\pi r^2$).