Short answer is yes. Moving air will cause noise, just like a turbo fan engine (very similar), and it is proportional to the fan speed (but is also dependent on the geometry and drive). Something NASA and Boeing and other aerospace companies have been trying to reduce for years, as are computer fan makers and HVAC system manufacturers. You need a good book on (fan) acoustics. Here are some references: 

See fig 18 http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19790006704.pdf 

http://www.nyb.com/Catalog/Letters/EL-12.pdf

https://www3.nd.edu/~atassi/Lectures/Boeing/Boeing0413_f.pdf

https://books.google.com/books?id=1x_RvffW-hcC&pg=PA836&lpg=PA836&dq=turbulent+noise+from+fan+engine&source=bl&ots=-07NO6PGzR&sig=G2tHaXcvlGz263bg9oV0HmqBWpI&hl=en&sa=X&ved=0ahUKEwj4xvqNiKjKAhUIbRQKHfMGBN8Q6AEIVTAI#v=onepage&q=turbulent%20noise%20from%20fan%20engine&f=false

https://engineering.purdue.edu/~propulsi/propulsion/jets/basics/noise.html

http://www.bu.edu/tech/support/research/visualization/gallery/fanacoustics/

Also, the frequency of the generated noise due to the excited force which may due to turbulent flow (pressure acoustic wave) is compared to the rotational speed frequency, if for only this frequency, there is a matching relation, and the noise will be purely due to turbulent flow. However ,if there is mismatching between the two frequencies ,further investigation can define the source of noise which may be due to mechanical reasons ,like imbalance in such case the noise will be in proportional to speed squared, by the same manner the analysis may be further investigated.

One has to refer the design of the fan. How it created eddies and therefore has to go through the handbook of fan design and acoustic. 

Yes. If you increase the speed, normally the noise will increase a lot.