It seems that wearing a hat is required, because the Shulchan Aruch says you need to wear a double covering. But some apparently-frum men don't wear a hat. How is this possible?
Many Yarmulkas are made from a top and bottom part which = 2 coverings. True Torah Judaism has absolutely nothing to do with the hat you wear.
No. The SA isn't the end all be all. The Biur HaGR"A (O.H. 8:6) even suggests that one can say berakhoth without a head covering. The idea that one cannot be an authentic representative of "true Torah Judaism" without a particular style of head-covering is absurd. This article is chock full of information on the topic of head-coverings.