Let $X$ be a g-genus closed surface. Then $X$ has pants decomposition.
Now consider $f:X\to R$ a morse function. Suppose critical points are ordered in increasing order and furthermore. Since $X$ is compact. There are finitely many critical points.
$\textbf{Q0:}$ First thing is there a way to concretely describe such a morse function. Normally there is a morse function on $P^3$. A generic $X$ is a smooth curve which can be embedded to $P^3$. Is the pullback of morse function on $P^3$ a morse function on $X$? $X\to P^3$ embedding usually cannot be described concretely as it involves cohomological argument and generic hyperplane projection. So my guess is that it is not possible to do so concretely.
$\textbf{Q1:}$ Is there a way to make sure that number of critical points can be reduced to $2g+2$? There should be $2g+2$ critical points if there is no extra discs.(Normally, each hole will introduce 2 critical points besides lowest and highest points.)
$\textbf{Q2:}$ Say $x,y$ are 2 neighboring critical points. How do I see that $x,y$ must be 1-handles in the opposite matching position?