For TOP Morse function a reference is the classical book of Kirby and Siebenmann "Foundational essays on topological manifolds, smoothings, and triangulations" (1977). The key point is to consider the local standard coordinate charts given by the Morse lemma in the smooth category, and use this to define the TOP Morse functions. These are strictly related to topological handlebody decompositions (so do not exist for non-smoothable topological 4-manifolds). In the PL category you can refer to the link of Daniel Moskovich or google "PL Morse function" (but the TOP approach is not likely to work).
Regarding the second part of the question, you can get such classification once you have proved TOP Morse functions exist! However, the techniques to prove in general that TOP Morse functions exist are typically high-dimensional (dim $\geq 6$). So for surfaces it is likely that proving the existence of TOP Morse functions is equivalent to proving existence of triangulations (which depends on the Schoenflies theorem).