Having a high willpower on a soldier increases the chance of psionic ability. If you put all your rookies through then you're not going to get many. If you wait until they've leveled up some and have avoided critical injuries (which give a corresponding loss in willpower), then the percentage will be much higher. I was seeing about 50% of my soldiers who had willpower 80+ testing successfully for psionic ability, yet only around 10% of my rookies (willpower 35-40).
The actual "Second Wave" options they are listing were already in the game, there was a user patch shortly after the game released which activated the hidden options so you could toggle them on/off each game you started. Some of them were a bit buggy at the time, but good to see they've enabled them for those of us who cba downloading 3rd party patches.