Greetings King of Ephyra!
I'll come back and mark your piece in full later, but some general points just to put your mind at ease:
1. You can offer a certain reading, but your piece is more likely to come across as persuasive than expository. You might find it helpful to examine the theory with/without reference to a few examples. Ultimately, you're trying to score points for context, so whilst an incredibly written existentialist piece would be really interesting to read, it may not be the easiest way to earn marks.
Skimming over your piece, I don't believe this is an issue. A more argumentative/politically minded approach would be risky, but existentialism and philosophy in general lends itself well to context, so you should be alright.
2. You don't
need to, and you could very easily score a 10 without doing this, but it would probably strengthen your piece to have some real-world examples just to demonstrate your points. Philosophy and theorisation is a good bridge, but some assessors would be looking for more varied evidence.
The exception to this is a hybrid piece; you could write something for a philosophy journal, (just put a name and date at the top, maybe change a few bits and pieces around) and then you're entitled to just operate within that area. Trouble is, links to the set text become a bit clunky.
Otherwise, some acknowledgement in your piece about where you're drawing your conclusions from might be necessary. I had a similar problem that was lit-centric. Almost every idea I had was from a novel of some kind, and so a lot of my essays had some variation of the phrase 'Although this is merely a fictional representation, many of the themes/ideas/conflicts are both inspired by, and applicable to reality.'
3. If it's something really well-known like, idk 'Hell is other people' or something, then it's probably unnecessary as most English teachers would recognise it as a quote. However, it's usually quick and easy to cite your sources internally, and the name 'Hemingway' alone should score you brownie points with the marker

For a general discussion of ideas you can move away from the original quote, especially when linking to WR, but I'd say the source itself can make a good starting point. Perhaps the lives of some of these great thinkers would make for some good external evidence?
Judging by your intro alone I'd say you'll be fine. Provided you can make connections in a sophisticated, but obvious way, there's little to stop you getting a high mark. If you are thinking about changing ideas or approaches at this stage, don't worry, there's plenty of time for fine-tuning.
Best of luck!