Complexity of Hopcroft-Karp
The correct running time is indeed probably $O((m+n)\sqrt{n})$. However, this is a mouthful, and the expression $O(m\sqrt{n})$ looks nicer and is also more succinct. In most cases, $m \geq n/2$, since otherwise there is an isolated vertex. In fact, by finding isolated vertices, we can improve the running time of the algorithm to $O(n + m\sqrt{n})$.

An alternative way of stating the running time would be $O(m\sqrt{n})$, assuming $m \geq n$. The case of small $m$ is not so interesting (since then there are isolated vertices which can be eliminated), and so it is usually assumed that $m = \Omega(n)$.