no its not cyclical unemployment - cyclical unemployment only occurs when a downturn in economic growth leads to a decrease in demand for labour, and thus increased unemployment
here, in the scenario you describe, the cause of the decreased demand for labour is the increase in the minimum wage, which decreases the attractiveness of labour as a resource in the production process and hence reduces demand for labour, leading to increased unemployment...
and, as a result, out of these:
– cyclical
– structural
– frictional
– seasonal
– underemployment
– hidden
– long term
the situation you have described does not come under any of them.
as for the second question, here's some good information that I found from my teacher's notes pertaining to the minimum wage and its relationship with unemployment:
"A decision by Fair Work Australia to increase award wages substantially in its Minimum Wage Decision (to improve the living standards of lower paid workers) might make it too expensive for some employers to keep all of their workers employed. Some employer groups claimed that the decision of Fair Work Australia’s Minimum Wage Panel to award a real wage increase to workers on award wages from July 2011, would result in higher unemployment because it would price some workers out of a job."
Hope this helps!