"You gain the special cards only when you experience a growth scene." I only got them once at my theoretical level 2 when I reached level 5. I guess there is a bug when there are multiple level ups stacked on top of each other, you don't get the special cards. I got to all 4 growth scenes and never got them again, because I used the cards at levels that were not dedicated to the growth sequence.
It doesn't matter when you use them. They are stored as points, which are displayed under your stats. It is written like "GP x (amount of points)". As long as this text is displayed, you can press K to level up a trait. The points work fine when I try it out in the exported version so I don't know what the issue may be like.