"Another question, does those draw and X active Champlain karma if you have exactly 4 cards in your hand?"
No, because you're not drawing it "into your hand".
"Also, is Antonie van Leeuwenhoek's karma negative? It reads as if it makes the achievement more expensive..."
No. Each card in hand counts as ten points "towards" the cost of achievements, not "added to" the cost of achievements.
"EDIT: on Bartholomew Roberts: it says ignore the age requirement, so the score requirement is still needed?"