I have come to believe that answering the questions I posted would be more enlightening if I try to provide an overview of the larger context that they are part of.
The literature treating and generalizing the topics mentioned in the post for surfaces is as extensive as it is interesting. The 1960's and 70's were times of very active research in this part of topology, and it still is today. Three wonderful resources are
Kirby & Siebenmann's book Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Milnor's paper Differential Topology Forty-six Years Later, and A. Ranicki's slides. I will refer to these as [KS], [Mil2011] and [Ran], respectively. Also, a word on notation: uniqueness will mean up to PL, DIFF or TOP homeomorphism, depending on the category at hand. Unless otherwise stated, words like manifold or surface will have general meaning (i.e. possibly with boundary and possibly non-compact). Finally, a list of references is included at the bottom.
- [DIFF & PL] (Strictly speaking PL and DIFF are not comparable. One uses the category PDIFF, which is equivalent to PL. However, this distinction is not normally made unless technicalities may require so.) Differentiable manifolds admit canonical PL structures. A differentiable manifold can be triangulated uniquely up to PL equivalence. S.S. Cairns first proved this result for compact $C^{1}$ manifolds, including those having a finite number of boundary components (See [Cai1934], [Cai1936]), although he generalized these results later (see [Cai1961]). J.H.C. Whitehead proved it for $C^{1}$ manifolds without boundary (see [Whi1940]), and J. Munkres finally included $C^{r}$ manifolds with boundary, $1\le r\le\infty$ (see [Mun1966] or Theorem 3.10.2 in [TL]).
A given PL structure on a topological manifold may have compatible differentiable structures that are inequivalent. That is, $$\mathrm{DIFF}\rightarrow \mathrm{PL}$$ is not injective. In [Mil1956] J. Milnor gave an example of a manifold PL-homeomorphic to the usual 7-dimensional sphere $S^{7}$, but not diffeomorphic to it. In fact, it is known that for $n\neq 4$ a topological $n$-sphere admits a unique PL structure (For $n\le 3$ see [Moi1977] or [TL], for $n\ge 5$ is due to Smale and can be found in [Sma1962]. The case $n=4$ is an open question). Therefore, the inequivalent differentiable structures that Milnor constructed in [Mil1956] are all compatible with the usual PL structure on $S^{7}$.
Even More, on the topological manifold $\mathbb{R}^{4}$ it is possible to define uncountably many inequivalent PL or differentiable structures. An excellent account of this exotic $\mathbb{R}^{4}$'s can be found in Chapter XIV of [Kir1989].
The functor above is also not surjective. That is, there are PL manifolds that do not admit a compatible differentiable structure. M. Kervaire gave such an example in [Ker1960]. Later, J. Ells and N.H. Kuiper (see [EK1961]), and I. Tamura (see [Tam1961]) gave examples in dimension 8, the lowest possible.
In dimensions 7 or less, PL manifolds always admit compatible differentiable structure, and in dimensions 6 or less this happens in an a unique way (See Theorem 2 in [Mil2011] and Theorem 3.10.8 and Problems 3.10.19-20 in [TL] for dimension up to three). In this sense, DIFF=PL for manifolds of dimension $n\leq 6$, which means that the number of inequivalent differentiable structures on a topological 4-sphere is also unknown.
The obstruction to finding a differentiable structure on a given PL manifold is called the Munkres-Hirsch-Mazur obstruction (see the last paragraph on [Mil2011]).
- [TOP & PL] $$\mathrm{PL}\rightarrow \mathrm{TOP}$$ is neither surjective nor injective. Indeed, there are topological manifolds, such as Freedman's E8 manifold, that do not admit a PL structure, or are even triangulable even if we allow non-PL triangulations. (A proof of this now follows from the proof of the 3-dimensional Poincaré conjecture, which implies that any triangulation of a 4-dimensional manifold is necessarily a PL-triangulation).
The exotic $\mathbb{R}^{4}$'s mentioned above provide an example of a topological manifold having uncountably many inequivalent PL structures. This disproves the manifold version of the Haupvermutung. The non-manifold version of the Haupvermutung was disproven by J. Milnor ([Mil1961]), who found two homeomorphic compact simplicial complexes that are not PL homeomorphic.
In dimension $3$ or less, the Haupvermutung is true (see Chapters 35 & 36 in [Moi1977] or Thurston/Levy's book). In this sense PL=TOP for manifolds of dimension $n\le 3$. Moreover, as mentioned earlier, except possibly for $n=4$ there is only one $n$-dimensional PL sphere.
The obstruction to finding a PL structure on a given topological manifold culminated with the resuts of Kirby and Siebenmann (the Kirby–Siebenmann class). (see [KS] and Theorem 1 in [Mil2011]).
- [TOP & DIFF] As John Klein points out in the comments, smoothing a topological manifold is in general a question formulated by first putting a combinatorial structure on the manifold (normally a handlebody structure or a PL structure). The examples of Kervaire, Ells & Kuiper and Tamura mentioned above yield topological manifolds having no differentiable structure. However, these are still PL manifolds.
More striking is the E8 manifold which, not being triangulable, cannot have a differentiable structure. It provides and example of a topolgical manifold of dimension four that admits neither PL nor differentiable structures.
The exotic $\mathbb{R}^{4}$'s mentioned earlier give an example of a topological manifold having uncountably many inequivalent differentiable structures.
In dimension 3 or less the results above yield DIFF=PL=TOP.
Coming back to surfaces I want to point out that Theorem 8.3 in [Moi1977] shows that every surface is triangulable. At the begining of the proof it is shown that triangulations and PL structures are equivalent structures on a surface. Moreover, Theorem 8.5 is the Haupvermutung for surfaces. Therefore, a complete classification of non-compact surfaces (with boundary) seems to have been achieved by the results contained and mentioned in Prishlyak and Mischenko's paper.
Finally, I want to point out that the result that two smooth surfaces are diffeomorphic iff they are homeomorphic is due to J. Munkre's and can be found in his dissertation Some Applications of Triangulation Theorems, U. of Michigan, 1955. The proof uses the triangulation theorems proven by E.E. Moise, who was Munkres' advisor.
REFERENCES:
The references [KS], [Mil2011] and [Ran] are listed (with links) right at the beginning, in the second paragraph above.
[Cai1934] Cairns, S. S., On the triangulation of regular loci, Ann. Math. (2) 35, 579-587 (1934). ZBL0012.03605.
[Cai1936] Cairns, S. S., Polyhedral approximations to regular loci., Ann. Math., Princeton, (2) 37, 409-415 (1936). ZBL62.0806.04.
[Cai1961] Cairns, S. S., A simple triangulation method for smooth manifolds, Bull. Am. Math. Soc. 67, 389-390 (1961). ZBL0192.29901.
[Ker1960] Kervaire, M.A., A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34, 257-270 (1960). ZBL0145.20304.
[Kir1989] Cairns, S. S., A simple triangulation method for smooth manifolds, Bull. Am. Math. Soc. 67, 389-390 (1961). ZBL0192.29901.
[EK1961] Eells, J.; Kuiper, N.H., Manifolds which are like projective planes, Publ. Math., Inst. Hautes 'Etud. Sci. 14, 181-222 (1962). ZBL0109.15701.
[Mil1956] _Milnor, J. W., On Manifolds Homeomorphic to the $7$-Sphere, Ann. Math. (2) 64, 399-405 (1956). ZBL0102.38103.
[Mil1961] _Milnor, J. W., Two complexes which are homeomorphic but combinatorially distinct, Ann. Math. (3) 74, 575-590 (1961). ZBL0102.38103.
[Moi1977] _Moise, E. E., Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics. 47. New York - Heidelberg - Berlin: Springer-Verlag. X, 262 p. (1977). ZBL0349.57001.
[Mun1960] _Munkres, J.R., Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. Math. (2) 72, 521-554 (1960). ZBL0108.18101.
[Mun1966] _Munkres, J.R., Elementary differential topology. Lectures given at Massachusetts Institute of Technology, Fall, 1961. Revised ed, Annals of Mathematics Studies. 54. Princeton, N.J.: Princeton University Press. XI, 112 p. (1966). ZBL0161.20201.
[Sma1962] _Smale, S., On the structure of manifolds, Am. J. Math. 84, 387-399 (1962). ZBL0109.41103.
[Tam1961] _Tamura, I., 8-manifolds admitting no differentiable structure, J. Math. Soc. Japan 13, 377-382 (1962). ZBL0109.16302.
[TL] _Thurston, W. P., Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy, Princeton Mathematical Series. 35. Princeton, NJ: Princeton University Press. x, 311 p. (1997). ZBL0873.57001.
[Whi1940] _Whitehead, J.H.C., On $C^1$-complexes, Ann. Math. (2) 41, 809-824 (1940). ZBL0025.09203.
Best Answer
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).