Drafting is done with draft boosters (and cubes, which are created by players for the purpose of drafting), never with other sealed products. You will never draft from a planeswalker deck, these are meant to be played in a constructed setting (everyone has their own deck already built).
Yes, people will get lucky in drafts (and sealed) and get stronger decks than other people, but the way it is fair is that it relies on that luck, no one is guaranteed if you sit 4th at the table you will open a planeswalker, and the rares in the pack are not always the best cards (or even useful cards, particularly if you spent the first two packs drafting different colors from it) The fairness comes from a draft or sealed being all luck and skill, you can't just go online, find the decklist that is doing the best and just buy your way into wins by buying those cards, you have to build it out of whatever luck gives you.
Lets look at your 3 questions
How can I get proper cards to build several decks, is it necessary to have cards from the Core Set in order to use cards from an Expansion Set, or do the Expansion Sets work standalone?
In 2015 Magic Origins was released as the last core set they ever intended to make. This wasn't popular and in 2018 they released Core Set 2019 again, but for years there were no core sets. Core sets are generally useful cards that can fit into many different places, while expansion sets are more streamlined, with specific strategies and archetypes being better supported. You can make a deck out of nothing but core set cards, and you can make a deck out of nothing but cards from a single expansion - usually you will make decks using the best of both worlds.
As for how to get the cards, you buy them. You can either buy cards from the secondary market, cards that someone has already opened and is selling, or buy packs. You also get to keep cards you get out of drafts, so that can also be a good way of finding cards you want. Generally buying specific cards is less expensive to get a deck, as there's never a guarantee you will get the card you want out of packs.
Does it even make sense to go for a Core Set display, if there is the Core Set Planeswalker Deck, which includes (I assume) the strongest cards of that set and guarantees Planeswalkers? Getting the display would most likely result in worse cards, maybe not even a single Planeswalker, or am I missing something? Maybe they are more common than I think?
You are talking about what is usually called a booster box. There is no 100% guarantee in these, they will not specifically contain the strongest cards in a set (and it's arguable that most of the time there isn't a strongest card in the set, when there is it tends to get banned like Oko, Thief of Crowns or Omnath, Locus of Creation were). This is just a box with 36 boosters in it, and if you buy boosters individually they are taken out of one of these boxes. The Planeswalker decks don't tend to be particularly strong decks, generally they are supposed to be decently balanced against each other, meant as a way for relatively new players to get into the game without excessively worrying about the small details that can go into competitive deck choices. They definitely are not built with the intent of being the strongest cards in the set.
How important are Planeswalkers for a proper deck, is a deck without inherently worse than a deck with or is it just a nice additional, which needs utilization?
Not at all, most decks have no planeswalkers in them. They tend to be flashy cards that get to do one thing before people kill them off or the effects are too minor to care about. Talking about the planeswalker decks, the planeswalkers in these tend to be underwhelming. They are different from the planeswalkers in the set, usually have a lower power-to-cost ratio, and rarely see play.
How many Planeswalkers appear in Core Set Displays?
If it was released before 2007, the answer is 0, since the card type didn't exist before Llorwyn. If it's a set after that, the answer is anywhere between 0 and over 36, depending on the set. Only War of the Spark guaranteed planeswalkers, 1 to 3 per pack, other sets can have none in a box or two in one pack (foil and regular).
To explain the cost/benefit of the planeswalkers:
These are the two copies of Oko that were in Eldraine. The first is the one that was in the preconstructed planeswaker deck (Note how the card is 309, it doesn't show that it's 309/269, showing that the card is outside the normal set cards), the second was the one you could get in packs. The planeswaker deck oko cost twice as much at 6 mana over the pack oko at 3. The abilities on the planeswalker deck Oko look flashy, but it takes 3 uses of the +1 to be able to get the ultimate ability off, 4 if you want Oko to survive it, and that's if no one kills him or hits him at all over those 4 or 5 turns. The pack Oko on the other hand can be played 2nd or 3rd turn, when people don't have much mana, is only 3 mana so you can play another one if the first one dies, and that +1 ability is all most people ever used, turning your own 1/1 creatures into 3/3s or turning your opponents big threats into plain creatures is a very powerful ability. Even if you wanted to use the pack Oko's ultimate ability, it takes one turn using his +2 to get there and he lives through it. This is pretty consistent over the other planeswalker deck walkers - they cost more and the effects look powerful, but the only really good one takes too long to be useful. Oko, the Trickster is too week and expensive for people to play, Oko, Thief of Crowns was so good he was banned 45 days after the set was released.