How are wind speeds typically distributed?
Yes, the most commonly used distribution for wind speed is the Weibull - at least when it comes to predicting wind speeds at prospective wind farm locations.

Yes, the parameters vary.

See, for example, the book "Wind Energy - The Facts", the output of an EU research programme (ISBN 9781844077106, also freely available online).

Or Seguro &amp; Lambert's paper Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis

As a starting point for estimation, it's usual to take $k=2$; this is equivalent to the Rayleigh distribution. However, the Weibull is more generalised, and often provides a better fit (i.e. $k$ isn't exactly 2).