Welcome to Azeroth!
These are good questions … (grits teeth) REALLY … good questions. 
I wish more beginners to WoW would think about them, because they are very important. Not easy to answer, tho.
- What WoW do I need to play?
In general, it will depend on what you want out of the game - what kind of experience you will enjoy. But when you say
that narrows it down fast. You want “Retail WoW”, which is now on its Dragonflight expansion and will soon launch The War Within expansion.
To join Retail endgame, you need to buy the latest expansion as well as buy gametime. You would pre-purchase The War Within expansion, which includes Dragonflight. You would then get one or more characters up to level 70 before TWW drops, seeing the dights, learning the ropes, and ready to start playing in TWW.
Why?
Classic is, at the moment, in museum status. Almost nobody is playing it. It is there for people to experience the original game (almost) as it was in 2005. I am very happy Blizzard is keeping that, but it’s not where you want to go.
Blizzard has been walking through releasing each of the original expansions in order. They started with Classic (aka Vanilla, aka World of Warcraft OG), then went on to The Burning Crusade, then Wrath of the Lich King, and now Cataclysm. I don’t expevt Cataclysm to be much of a success; it wasn’t a popular expansion, it changed the game A LOT, and most people would call the three original versions the “Classic Era”.
While there is an endgame in Cataclysm, it will be much narrower, and frankly not as rich, as the current enfgame in Retail WoW.
And that’s why you want Retail WoW.
1a. What’s with the story?
/sigh
Yeah. Yeah. That’s a thing. It’s a big difference in philosophy between WoW and FFxiv.
Warcraft’s story starts at the creation of the Universe, and does not center on the player (though they have made a - IMHO - bad change in recent expansions towards making the player an Important Character in the expansion).
Here is a Timeline for the Warcraft Universe. I know it’s long, but worth becoming familiar with or referring back to:
It spans multiple games and books, and not all of the story is in WoW itself. Quite a lot (far too much, IMO) cannot be found in-game, but only in the books.
The problem is that, having been developed and written over 20 years, the total story of WoW is HUGE!
There is a - wait for it! - 514 video series of the lore of WoW here
https://www.youtube.com/playlist?list=PL6037698E84060287
Though this 40 minute video https://youtu.be/RKsNKZi5Ct8 hits the highlights up to two expansions ago.
Honestly, I think the best way to approach story is to consider yourself thrown into a world that has a lot of history, and just worry about surviving first, and understanding the backstory as and when it becomes relevant to you. Isn’t that how we all live in The Real World anyway?
As a new player, you will be started in the Battle for Azeroth timeline. Depending on whether you play Alliance or Horde, you will be introduced to characters and situations that will give you a general feeling forsome of the enduring characters on each side. And then you will start Dragonflight, which has minimal ties to the long-term story. Players seem to regard it as a palate-cleanser between a truly rotten story in the previous expansion Shadowlands and the upcoming Trilogy of expansions starting with The War Within, during which we will aid the Titan Azeroth who is about to be born within our planet.
- What server do I play on?
As Alliance, choose a high-population Alliance server. As Horde, choose a high-population Horde server.
The difference is that you will see more people around you in the world that you can talk to, group with, join guilds with, and trade with. But that difference has been decreased a lot, and will decrease more with the arrival of cross-realm guilds in TWW.
Here is an indication of the relative size of servers, and the split between Alliance and Horde
and you can sort by faction population.
Thisis MUCH LESS important than it used to be. There is already cross-realm trading and limited cross-realm Auctions in Dragonflight, and there will be cross-realm guilds in TWW. You can join groups with mixed realmd and mixed factions.
Honestly, if I were starting now, I don’t know what I would choose. Up to Dragonflight, I would say to choose one of the Top Five realms for the faction, but I don’t know how the changes in TWW will affect populations. I think it would push my choices down the list somewhat.
The top two or three realms for each faction can have overcrowding issues like lag, and realm disconnects, for weeks at the start of a new expansion. I would usually say that’s worth it for the increased pool of players and economy, but depending how players spread out in response to the new freedoms in TWW, the advantages of a very high-pop realm will be limited, So maybe I’d say look at numbers 4-9 rather than 1-5 for your faction, but honestly nobody knows how it’s going to play out.
This probably doesn’t alnswer everything, and may confuse your further, but it getting massively log, so do please ask any followup questions you have. 