Charge, Size, and Movement in Aqeuous Ions as Electrolytes
Would a pair of aqueous $+2$ and $−2$ ions necessarily conduct electricity better than the same concentration of $+1$ and $−1$ ions?


Not necessarily. Taking the concentration of 0.001 aqueous solution of $\ce{KOH}$ and $\ce{MgSO4}$ at $18$ degree Celsius, we find that the current conducted by $\ce{KOH}$ is $2.34 \space mA$ while the current conducted by $\ce{MgSO4}$ is $2.00 \space mA$. Why did this happen?

It happened because of the mass. And therefore here comes the answer to your second question.


  Would more massive ions not conduct as well as less massive ions because they cannot travel as rapidly?


Yup, more massive ion means less current. It's because of simple equation (and I hope you are aware of these physics equations)$$qV=\dfrac{mv^2}{2}$$

Here $q$ is the charge on the ion. $V$ is potential difference between both the solutions. $m$ is the mass of the ion. And $v$ is the drift velocity. We know that $qV$ is constant. So we get $$v \propto \dfrac{1}{\sqrt{m}}$$ We can easily analyse that more the mass we have, less the speed we get.


  If ions are moving, why don't they ever achieve the most stable position and stop moving?


They do achieve the most stable position and that's why batteries die.