Complexity class of an algorithm
No, $\sqrt{n}$ increases far faster than $O(1)$, and $n^{\sqrt{n}}$ grows far faster than $n^{O(1)}$.  No, it certainly does not have the same runtime.  See Sorting functions by asymptotic growth.

There may be no predefined complexity class; the complexity class is the class of all algorithms who run in time $n^{O(\sqrt{n} \log n)}$, and there's probably not much more to say.