Hi Gabriel, let me see if I can answer your questions...
1) Yes, that sounds a bit odd, although it might be because it's not heard very often. When I hear this, I picture someone trying to pick up a couch or TV by putting his/her hands on the bottom part and then lifting it up, but we usually just stick with "lift" as the verb for that.
2) Yes, you can say both. "I've never gone whitewater rafting" is a perfect sentence. In this case, "whitewater rafting" is an activity that you do somewhere farther away from where you are, so "to go whitewater rafting" makes perfect sense.
3) Both "put" and "set" can be used in this case. You are correct both ways. Although if you want to be more specific, "put" is often suggests more permanence/longer period of time whereas "set" suggests something more temporary, but most of the time, English speakers don't worry about this and use both words interchangeably.
4) I've never heard "take (something) off a drip," so I don't know what that means. "Take a medicine through an IV" (notice the "an") is correct, as is "take out the IV."
I hope I've helped. Please feel free to ask any questions you may still have.