1. My opinion is that it sounds equally natural with or without the indefinite article (which is "an" not "the", by the way). If you were the only invigilator, then the definite article ("the") also sounds correct. My preference would be "for", but "at" is not incorrect.
2. "At/for" here is no more than individual preference. I would not say one is more correct than the other. "The external examiner" is definitely the correct form, however. Without "the", it is not incorrect, but sounds unnatural.
3. Yes. In this case, "the" is mandatory. We are referring to a particular examiner.
4. Exactly the same as 3.
Student or Learner