Usually we refer to "strengths" and "weaknesses" in the plural, so it would be "are" not matter what.
You could, I guess, say "What is your greatest strength and greatest weakness" with the second "what is your" implied after the "and" or because it sounds like your greatest strength and weakness are a pair of things considered together.
You could also say "What are your greatest strength and your greatest weakness?" thinking of them as two different things.
My recommendation, though, for your first question is to make them both plural.
Student or Learner