Is it safe to pass a natural gas hose under a wooden step, and to keep it pressurized?
Before we get in to the answers to your specific questions, you first need to read the instructions that come with the hose and whatever you hook it up to. For example, the hose may not be rated for continuous outdoor UV exposure. And your grill probably says to turn off the gas at the source; the burner valves are not designed to hold up against supply pressure indefinitely.


  Is it safe to take a flexible natural gas hose such as the one pictured and pass it under a patio's wooden stairs to reach a BBQ? Can it simply lie on the concrete under the patio?



That should be fine. Just make sure it's not easily stepped on, or rubbing against an edge. And don't let water freeze on it.



  Can you ever leave a hose such as the one pictured pressurized overnight, or must you always turn off the incoming gas, rather than simply turning off at the BBQ?



No. You need to turn the hose off when it's not in use. If this were connected to a cylinder it would be less of an issue since the cylinder has a fixed volume of gas. But since this is hooked up to a gas supply line, it could be days or weeks before a leak is noticed, potentially creating a huge pocket of trapped gas.



  can such a hose be left outside over winter?



Check the ratings on the hose.