Even without minimum wage there is a price point where I'm better off not taking the job and trying to scrape out a living in a more ad hoc way.

Unemployment existed before those; and whether minimum wages increase unemployment should be open to empirical study - anyone got any data?

Empirical study is a tad tricky in a field where experimentation would probably be considered wildly unethical and you wouldn't be able to set up a control anyway.

While presence of controls is admirable, it's still reasonable to simply look at countries that implemented a minimum wage and ask, what happened to the unemployment rate?

At the very least one would note that the effects don't seem to be predictable via the basic supply & demand theories.