From robjohn's answer to a related question, I found a workaround that solved the main issue and the first of the secondary issues. Using the code $\quad$ \sqrt[\raise{7pt}\Large3]{\frac ab} $\quad$ enclosed in \$\$s, we get $$\sqrt[\raise{7pt}\Large3]{\dfrac ab}.$$

From robjohn's answer to a related question, I found a workaround that solved the main issue and the first of the secondary issues. Using $$\sqrt[\raise{7pt}\Large3]{\frac ab}$$, we get $$\sqrt[\raise{7pt}\Large3]{\frac ab}.$$