can Existential Questions be discussed analytically?

Here's some:

Why is there something rather than nothing? - Heidegger
What must be the Being of a man if he is to be responsible for and capable of bad faith? - Sartre
Is the past more necessary than the future? - Kierkegaard

What must be the Being of a man if he is to be responsible for and capable of bad faith? - Sartre

Sartre's question "What must be the Being of a man if he is to be responsible for and capable of bad faith" is a very badly worded question. Let us try to rephrase and analyse the question:
"What must be the Being of a man" may be rephrased as <What is the dilemma, contradiction or paradox of a man> and let us see this question with the second part of the sentence: "if he is to be responsible for and capable of bad faith?"

Does replacing "Being" with "dilemma/paradox" maintain the meaning of the question?

Here's some:

Why is there something rather than nothing? - Heidegger

What must be the Being of a man if he is to be responsible for and capable of bad faith? - Sartre

Is the past more necessary than the future? - Kierkegaard

....does [it] make contingent things that DO come into "real existence" more necessary (or come closer to necessity) than those that do not come into existence but stay in conceptual existence?

Thank you Victor Eremita for providing us debating points but since you put the above question I expect you to come up with the answer please.

[ 1 ] Does the Possible become more Necessary by becoming Actual"? - Kierkegaard
There are almost endless "flip sides" for mystics and mysticism fans. Therefore, whatever an answer you reason to a specific question, mystics usually come up with another answer to prove that you have been wrong since the keys to all questions are kept within the hand of those mystics, gurus, rabbis etc who may change the rules of the argument to suit their cause. Therefore, mystical and theological arguments should never enter philosophy pages. It is destruction and waste of time.

As a straightforward logical thinking individual, this is not a sentence with meaning since the words "possible", "necessary" and "actual" have meaning only within a defined concept, and this sentence does not give us a clue of what the concept is. However, I understand that Kierkegaard is making a version of the ontological argument for the existence of God which should have been understood to be useless by now.

[ 2 ] Does truth always involve communication? - Arendt
I am not quite sure of the existentialist nature of this question but my answer is: The truth of the matter is independent of the observer. Therefore, answer to this question is NO. However, communication is the means for the realisation of such matters by the observer(s).

[ 3 ] Must I always reason from existence, not toward existence? - Kierkegaard
This is a question, I believe, that says a lot about the Man who starts his reasoning on a priory premise that "God exists" and shows his doubt and perhaps resentment that this priory premise may need some reasoning before accepting it as a true premise. To do that we need to go to Descartes' "I think therefore I exist".There is no "a priory" premise here. Why then do we need to accept "God exists" as a priory premise and build everything on this questionable foundation? This is a theological question, not philosophical.

[ 4 ] Does identity form social gender? - Beauvoir
Identity or "self" together with "freedom" are the main components of the existentialist philosophy and "self" obviously connected to "gender" which reflects the phyisical and mental state of the self in a certain way. But I am not quite sure what "social gender" means.

Existentialism could have been real fun if only we were not distracted with non-existentialist, non-logical and non-philosophical questions.

For Arendt, truth does involve communication in the political arena; one defines oneself through communication (indirect and direct) from Others.

And for SK, proving existence from attributes like 'thinking' is like proving all thinking things exist. Kierkegaard would rather demonstrate existence, than futilely proving existence.

Beauvoir asks whether identity alone is enough to create gender; or that it is the case that "one is not born a woman, one becomes woman"

Part of the allure is that Existentialism isn't just philosophy, it's also literary, political, psychological and in some versions, theological.

Much confusion can arise from the ambiguity of words.

Surely existence necessarily follows from any attribute. Where an attribute is present, something must possess it; and whatever possesses it must exist (by virtue of the meaning of the word 'exist'). How could thinking exist, but not the thing doing the thinking?

Furthermore, if 'essence' (another vague word!) means 'the possession of attributes', then it is difficult to see how existence can precede it, since everything must have some attribute, otherwise it would be nothing. The definition of 'essence' needs to be precisely agreed before it can be usefully discussed.

Again, it entirely depends on what one means by 'gender' and 'woman'.

Literary appeal is one thing; but if it is to be judged on a political, psychological or theological basis it ought to be philosophically sound, i.e. clear, coherent and rational. It is therefore a good idea to discuss it analytically, in order to correct any inconsistencies, vagueness or unfounded assertions. There is no reason why philosophy cannot be both rigorous and stylish!

"x" exists
"x" has attributes "walks", "bald head", "average frame", "says his name is Jean Luc Picard" ...etc.
---------
Therefore "x" is Jean Luc Picard

Do we say Picard exists? Yes, but we reasoned FROM existence, not towards it. "x" exists first, then we show how "x" is Picard.

"If one wanted to demonstrate Napoleon's existence from Napoleon's works, would it not be most curious, since his existence certainly explain the works but the works do not demonstrate his existence unless I have already in advance intepreted the word "his" in such a way as to have assumed that he exists. But Napoleon is only an individual and to that extent, there is no absolute relation between him and his works-thus someone else could have done the same works. Perhaps this is why I cannot reason from the works to existence. If I call the works Napoleon's works, then the proof is superfluous, since I've already mentioned his name. If I ignore this, I can never demonstrate from the works, that they are Napleon's but demonstrate purely ideally, that such works are the works of a great general, etc."

If you see "x" doing philosophy, Victor Eremita's existence can explain "x" doing philosophy (it could be Victor Eremita doing philosophy), but you can't logically prove "x" doing philosophy entails Victor Eremita's existence; it could be VideCorSpoon doing philosophy or Justin doing philosophy. But what you can do is prove existence purely ideally that "x" doing philosopy exists, But you are not logically proving that just because "x" is doing philosophy that you prove a particular individual's existence! Here's where the psychological aspect kicks in.... it looks like Victor Eremita, speaks like Eremita, walks like Eremita therefore it is Victor Eremita, dure... who else could we be talking about?! But that's psychological, not logical.