1) These words are synonyms, but can convey varying degrees. Important is a gradeable adjective, so something can be "very important." However, crucial is ungradeable. It's already at the top of the scale of importance, so you can't say "very crucial." They all mean basically the same thing, but would be used differently simply to connote varying degrees of importants, or to lend emphasis within a sentence.
2) Your sentences are grammatically correct, but these are not likely to be the words a native speaker would choose in these instances, particularly in the second sentence. :)
3) Not sure on this one... are you just looking for terms that mean what you've described? "Always keeping in my wits" is not a proper term. "Keep my wits about me" would make more sense. However, I'm not entirely sure what you're asking here.
Student or Learner