1) Do you say "Death by drug overdose/asphyxia/cancer" or "Death through drug overdose/asphyxia/cancer" or "Death from overdose/asphyxia/cancer"?
--We would use "by" in those phrases. We would more likely say, "He or she died of an overdose, asphyxiation, or cancer," and not, "It was death by overdose or asphyziation or cancer."
2) What does "Set them up with another appointment tomorrow" mean?
--It means "Schedule an appointment tomorrow."
3) If you know everything about an artist's work, could you say something like: "I've been following/keeping up with their work since I was a kid"? Or does it sound odd--
--Those phrases are perfectly fine.
4) Do you say "They made a foray on the enemy/the enemy's territory" or "They made a foray into the enemy/the enemy's territory"... "Swoop on or swoop in the enemy's territory"?
--Yes, you can say: "...made a foray into enemy territory," and it sounds fine.
What about "Made a incursion/drive/onslaught on the enemy's territory"? Can you say any of these?
--Those all sound odd. Can you just say, "invaded enemy territory?"