Reference needed: lower bound on sample size for determining which side of coin is biased with high probability
Yes.  You need approximately $\sim 1/|p - 1/2|^2$ observations (that many are necessary and sufficient), assuming $p$ is close to $1/2$.  I've omitted constant factors, but the constant is small.  The exact constant will depend on the degree of confidence you want, but it will vary as something like $\log (1/\delta)$ (or lower): $\delta$ drops exponentially fast in the number of observations.

For more details and analysis, see https://cstheory.stackexchange.com/q/22328/5038.  You might also be interested in https://cstheory.stackexchange.com/a/30938/5038.