Run-time of Sorting Algorithm
This is just Bubble-Sort in disguise. Each time, you get the biggest element pushed to the end of the array, then you sort the remaining of the array.

Your recurrence is correct.

An upper bound would be $T(n) = O(2^n)$.

Proof: For large enough $n$, we have $T(n)\leq 2^n$. This is the base case for induction. Induction hypothesis is $T(n)\leq 2^n$. So, we have $$2\sqrt{2^n}+\frac{2^n}2+O(1)\leq 2^n$$.