How should the client enter this market?

As you describe it, this is the longest version of this case format; normally too long for an oral interview, so it's super important to listen whether the question is about if (the client should enter the market) or how (to enter).

Let me answer your direct question 2) first, the how. This is the general "market entry strategy" case. For this strategy, you need the following:

1. Product strategy -- what are you going to sell? Your existing product as is, a high-spec or low spec version of it, or something else? (Hint: It needs to be at least adjacent, either new product for same customers or same product for new customers, otherwise it makes no sense; steel mills are rarely becoming tech cos. Compare the Ansoff-Matrix)
2. Geographical region -- where are you going to sell? Needs to be defined, often already part of the case question itself
3. Distribution strategy -- how are you going to sell or who is selling your product? Which channel? Own shops, specialty shops, big box retail, online, partnerships, etc. Depends on the product/service, for services or B2B it's Sales (inside sales, field sales, sales partners) instead of channel.
4. Promotion strategy -- how will customers know about your product? Often forgotten even by consultants :), but highly important. Marketing.
5. Pricing strategy -- what will you charge? In essence, you figure out steps 1-4, and then figure out max price point "allowed by" competition given quality of the client's offering.

Few comments:

• Customer (segment) strategy is missing. I tend not to include it, as to me it's above the framework, you need to figure it out before doing all 5 steps. Take into consideration if necessary.
• Competition - to me it's not a separate step as it's nothing you do or decide. I would bring it up as part of pricing. As consultants, you advise the client, and case result is rarely "competition is too strong, let's pack our bags and go home". One can get into any market with an attractive offer, and in 90% of cases it depends on the client's capabilities more than competition. Should be evaluated when testing market attractiveness.
• Some of these steps may be a given, i.e., part of the case question (e.g., "client wants to expand to the UK", or "Chinese solar company wants to expand to U.S."). Don't question the question, solve for the remaining variables.
• Do not overestimate clients' capabilities, clients are typically multi-billion-$ companies with lots of bureaucracy. Changing all 5 points from above won't work in practice. (New product in new region with new ...)
• The power of the status quo. One trick for you is to ask how the client is selling their current products right now in their home market. Chances are, approach in the new market will be kind of similar - if they sell detergent in supermarkets now, it's highly unlikely that selling online will be the solution in another market.
• If this sounds awfully similar to the 4P - it's because it's EXACTLY the same. Compare my answer here https://www.preplounge.com/de/consulting-fragen/mece-ways-to-grow-market-share-205 (Over time I found there are only 6 major case approaches that kill 95% of all case questions, so would be strange if this would be completely different :)

Basically memorize the 5 steps, and you're good!

Regarding your 1), you're completely correct, but I would prioritize it way more. Key points for me would be:

• (Long term) growth -- trumps everything. Power of compounding interest; even if current market size is small or (current) profitability is low
• Client capabilities - what does the client bring to the table? Should be either product capability (they can do X already so not hard to do Y, or, major R&D invention or something), or customer access (weaker, but still good -> cross-selling). Here I would compare it to the competition, as cababilities only make sense on a relative basis anyways.
• Your other points are all important, but if it's a really fast growing market and the client can play in it, you can directly go to step 2) and figure out the details.

I should shorten this answer, but rather get some sleep instead... Hope it still helps, good luck with your interviews!! :)