Can an Autogyro with wings survive a rotor mechanical failure? Will it be safer?
That is a very broad question and covers many topics, but I will try to answer as best I can and not miss anything.

A quick note on the wings.  Adding wings as a safety aspect to the aircraft kinda defeats the purpose of an autogyro.  

Would it survive to a rotor destruction? The likelihood of an autogyro surviving rotor destruction is extremely remote. Best case scenario would be the entire rotor assembly detaches from the mast and flies away from the aircraft.  Worst case scenario only 1 or 2 blades fly off. (Honestly would be more likely to happen IF rotor destruction were to take place.)  In this situation the imbalance in the rotor would be so violent that surviving, let alone actually flying, an autogyro in such a configuration would be truly "Hand of God" type miracle.  If you want to see something similar try removing a blade or two from a fan and you will see this imbalance in action.

Taking the best case scenario could the pilot make an emergency landing? Yes, but their options would be limited, assuming you can recover AND get into the best glide configuration by 5,000' AGL you would likely have less than 8.3 minutes to find a field and get ready before landing. Lower altitudes will provide even less time before contacting the ground.  (This number is only approximate for a 9.3:1 glide ratio and it would be unlikely for wings on an autogyro to even do that well.)

Is it safe to build an autogyro with wings? Yes, as your picture shows it does work.  A few companies today, such as Carter Aviation Technologies, use that design albeit with much shorter wings designed to work in conjunction with the rotor.

Does it have an increased operational cost? Possibly.  Those smaller wings mentioned in the paragraph above actually increase the efficiency and reduce the operational cost.  The problem is that those types of wings are almost entirely useless without the rotor.  Wings big enough to provide a decent glide ratio and time for the pilot to react to rotor destruction would end up providing more drag and require a larger engine to overcome it thereby increasing the operational cost.

As far as turbulence goes, you could see more turbulence with larger wings but with small wings like on the CarterCopter, the additional turbulence added to the aircraft because of the stiff wings would be minimal.