From what I've read, some economists thought Germany was running into poverty problems because of their lack of a minimum wage? I'm not sure of the details. I'm not entirely sure having different minimum wages by industry is a good idea either, because unless done very carefully it adds an exogenous incentive toward certain jobs that can't be good for a free market.
Yeah. I think that there should be definitely be a minimum wage, which is pegged to some sort of measure of cost of living. Trade unions can negotiate anything on top of that for their respective professions, but if you're not represented by a powerful union that shouldn't consign you to poverty. This article http://www.thelocal.de/20131121/germany-to-introduce-minimum-wage has some examples of the really low wages that German workers have had to deal with - office workers receiving €1.37 an hour, delivery drivers receiving €1.55 an hour in Mecklenburg-Vorpommern. This is pretty dreadful. The SPD wants to make the minimum wage €8.50 an hour.