Depends on the bbq technique; many smokers include a water pan to keep the humidity high (and also to act as a thermal reservoir).
Yes, propane and natural gas have lots of hydrogens to combine with atmospheric oxygen to make water.
Well doesn't burning wood also generate water? In addition, doesn't seasoned firewood consist of 10-20% straight up water that boils when the log is burned? Propane doesn't contain water.
Water is one of the by-products of the combustion reaction, regardless of what fuel is used.
Not unless the fuel contains only carbon - as charcoal does.