That's a ton of questions right there :)
Two things are most important for all of these questions/cases, which by the way also apply to most other cases:
I) Answer the question. That will put you ahead of ~60% of other applicants (I'm serious). If the question is about "profit", your answer at the very end should be a number (with a unit!) for the profit amount. One of my favorite questions in the personal fit part of the interview starts with "Name 3 adjectives that..." -- you wouldn't believe how many candidates fail to name 3 (mostly they just come up with 2, but I have gotten 5), or fail to name adjectives. While not getting it 100% right doesn't have to be a dealbreaker per se, there is really no need to make this mistake. And this is what consulting in general is about: Answering specific questions with specific answers.
For this case, there's very little context about the industry or anything, so you could structure it as follows (just one possible option off the top of my head):
- Define the goal. The goal is to convince the CEO to hire your firm for a project. Repeating the goal of the case shows you're working in the right direction, keeps you from getting off track, serves as step 1 out of 3 (only 2 steps remaining, you always need 3 steps for ANYTHING in consulting :), and brings 45 seconds thinking time.
- Define requirements to reach the goal. If you want to sell a project, requirement is that you offer value to the CEO (greater than the consulting fees). So, what would the CEO pay for? To get one of his most important problems solved. So you need to find out what they are. Possible questions (after small talk) to find out are "What keeps you up at night?" (a bit overused), "If you had one free wish/a genie came to you to solve one of your problems, which would it be?" (better maybe even: "take one of your issues off your plate") or "What are the most difficult problems in your industry right now - do they also apply to you?", or, the scare tactic, "Where is your competition better than you?". That should get the CEO talking, and you can ask further and more detailed questions to understand his problems. (Maybe there is a better term than requirements, but you get the idea).
- Offer proposal. Once you understand the problems the CEO is facing, you would structure a proposal how to help him, answering WHAT you can do, and WHY you can do it. "What" is basically throwing expertise and (consulting!) resources at the problem, "Why" has to convey confidence that you're the right person to do it, appealing to the CEO's emotions. It's basically a variation of "we've done it before" (in an adjacent industry, etc.).
So, if the question is asked the way it is, my approach in the interview would be to explain the structure first (basically, the three steps in bold), and then detail them out. Alternatively, detail 1) out to cash in the thinking time, and then show the structure for 2) and 3), and then afterwards detail 2) and 3) out. Then, at the end, ANSWER THE QUESTION! People call this "summarizing", but it's not, it's basically showing that you know what the problem was and to give the answer. In this case the answer to "how would you convince the CEO to hire your firm for a project", aka, the goal from 1). Why am I stressing this? The answer to this is not 3) the proposal -- the case question is about an overall approach to the CEO talk, so the answer needs to include all three steps.
By the way, the question would be even more difficult as a roleplay, because then you need to do all of this in your head, and package it into a nice and friendly (= read "non-awkward") conversation.
II) Show confidence. This is the second most important thing in all interviews. It subsumes:
- Being in charge - the interviewee leads the case.
- Posture/sitting up straight.
- Clear, confident voice.
- Being "natural" -> a goldilocks state of perfect moderation (not too loud, not too quiet. Not talking not too much, not too little. Not too fast, not too slow. ... )
- Be confident enough to listen to the interviewer, in most cases, she or he will try to be helpful (but don't "beg" for guidance).
Why is this especially important in consulting? Most of your clients are C-level types, typically alpha-females and -males. Many are used to having staff who look up to them or are scared to speak out. One of the roles in consulting is to speak to these executives as an equal, being a true partner and guide. And if you show submissive behavior, they don't take you seriously. And for sure you don't sell projects then. (Ok, this is a bit of an overgeneralization, but you get the general message.)
Since you mention "panicking" in your question, one trick I learned prior to my interviews years ago to help me was this:
Think about a similar situation where you felt confident and in complete control. For example, a talk you gave in class about a topic you knew in and out, or a presentation you aced. Think about how you felt and behaved in this situation. (Posture, breathing, voice, ...). Get yourself in that state, and try to "save" it into a specific gesture -- my gesture was pinching my left thumb with my right thumb & index finger. Repeat and practice that 3 times a day for 5 mins each. Once you're in the interview and feel panicked/stressed, recall that feeling by secretly performing the gesture. You really need to physically practice this until it becomes subconscious, otherwise it won't work -- it's of no use that you get the concept intellectually! It actually really helped me, despite not being a fan of esoteric stuff in general, but hey, whatever works! ;)
Sorry for the long answer, but I didn't have the time to write a short one :) (Pascal)
All the best for your interviews!