Firstly, with regard to Heinrich Cartwright's character ability. It states that whenever you draw one or more crads during your turn you may draw one extra and then discard one. The way I read this is that this applies when drawing cards through card effects, and also when drawing up at the end of your turn.
Correct. You are technically drawing your hand back up during your turn. I occurs at the end of your turn inclusively, as opposed to after your turn or in between.
What happens with the dark secret card if no shocking discovery can be drawn?
It is simply destroyed with no other effect.