Ivo B:
If I had to choose between being right or being happy I would always go for the first one.
I like how you state this. Lots of people seem to share your point of view, but most of them don't realize it or won't admit it. At the risk of sounding obvious, I feel I should mention that the Buddhist path is about one thing and one thing only: cessation of suffering. If you are not actually interested in this as much as you are in being right, then this is probably not the path for you.
Ivo B:
Those are fundamental questions of our existence and if in the process of getting enlightened those questions lose meaning I'd sure like to know the answers as to why did they lose it.
It's sort of like this: Say, for example, you are on an Internet forum and someone starts telling you how you are extremely bad at math. He claims that you obviously don't even know how to add or subtract, much less multiply or divide, and he wonders how you even survive, whether you could possibly have a bank account, etc. As a result of this, you may wonder, "Why is this person saying these things about me?" This questions may seem somewhat important to you. A few posts later, it suddenly turns out that the poster had your username confused with someone else's. He never meant to address you in the first place. Suddenly the question, "Why does he think I'm so bad at math?" loses pretty much all its meaning, simply because you now realize he wasn't even thinking about you when he said those things, but in reality he was thinking about someone who has nothing to do with you.
Before SE, you may wonder, "Who are we?" "What are we?" "What is going on?" and in your case apparently fist and foremost: "Why do we exist?"
You don't wonder why there is a pink elephant in your living room, correct? That's because there is no pink elephant in your living room. This fact is something you believe because it is in accordance with your direct experience, therefore the corresponding question is unimportant to you.
If I told you you didn't exist, then you wouldn't wonder anymore why you exist. ... except: The only flaw with this is, of course, that you wouldn't believe me that you don't exist, because my assertion seems totally at odds with your direct experience. Enlightenment will fix this. It will show you directly that you don't exist.
You may respond: "But my thoughts exist. My questions exist. My body exists." Yes, those thoughts and the other things exist, but in a different way than you thought before SE. Now that you understand the nature of thoughts, body, etc, much better, you will ask a different series of Why questions, one that the Buddha has answered in what is called dependent origination, nidana, or paticca samuppada:
Why is there suffering, old age, death, sorrow, lamentation? — Because of birth.
Why is there birth? — Because of becoming.
Why is there becoming? — Because of clinging.
Why is there clinging? — Because of craving.
Why is there craving? — Because of feeling.
Why is there feeling? — Because of contact.
Why is there contact? — Because of the six sense bases.
Why are there the six sense bases? — Because of name-and-form.
Why is there name-and-form? — Because of consciousness.
Why is there consciousness? — Because of volitional formations.
Why are there volitional formations? — Because of ignorance.
(There is also a slightly different version of this in the Maha Nidana sutta.)
Ignorance in this context is, of course, not the "lack of answers to questions you might have," but it is simply "not seeing things as they really are." You fix this by learning how to see things as they really are, not by finding, hearing, reading, or memorizing answers to questions.