Why would you expect to find a collision in a hash function after approximately $\sqrt{n}$ hashes? I can't get an intuitive understanding of why it's $2^{(\frac{n}{2})}$ and not $2^n$, where $n$ is the number of bits of which the key consists.
A collision is between two values. If you take a random pair of values you get a 1/2n chance of having a collision.

With 2n/2 values you have about 2n-1 pairs, so you could expect about 1/2 chance of collision.

(That's just the "intuitive way" of thinking about it; in practice, there are mathematical details.)