It's a result of lowdimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. Milnor's 7spheres give nice counterexamples to this result in dimension 7, and exotic $\mathbb{R}^4$'s give nice counterexamples in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6? And, if false, what are some classic counterexamples, and do stronger constraints  say compactness or closedness  happen to make it true?

It is false in dimension 5 and 6. Spheres happen to be standard, but some other (compact and closed) manifolds happen to admit different smooth (and PL) structures. Simple example are tori. For example, $\mathbb T^5$ admits 3 different PL structures that give rise to 3 different differentiable structures. See, e.g., Hsiang, Shaneson "Fake tori" or Wall's book on surgery. 


Any PLmanifold of dimension $\le 7$ is smoothable, and the smooth structure is unique in dimensions $5,6$. See e.g. remark 6.7 in Rudyak's paper for details. EDIT: To explain the above, the smooth structures on a PL manifold $M$ of dimension $\ge 5$ are in 11 correspondence with $[M, PL/O]$, homotopy classes of maps from $M$ to the space $PL/O$, which is $6$connected. This implies the claim in the previous paragraph. Similarly, PL structures on a topological manifold $M$ of dimension $\ge 5$ are in 11 correspondnece with $[M,TOP/PL]$, and $TOP/PL$ is $K(\mathbb Z_2,3)$. Thus $[M,TOP/PL]$ is simply $H^3(M;\mathbb Z_2)$, the third cohomology group with $\mathbb Z_2$ coefficients, and if $H^3(M;\mathbb Z_2)$ is nonzero, then $M$ admits more than one PL structure. See MadsenMilgram "Classifying spaces for surgery and cobordism of manifolds". 


AkhmedovD. Park paper on exotic $\Bbb CP^{2} \# k\overline{\Bbb CP^2}$ for $k \geq 2$ long predate FintushelStern's preprint. Their papers written in 2007 and published in Inven. Math. Also, AkhmedovPark recently posted a preprint on exotic $S^2 \times S^2$. I am wondering if their method extends to $\Bbb CP^{2} \# \overline{\Bbb CP^{2}}$ or $\Bbb CP^2$? 

