This question gets a bit tricky to answer... Starting from scratch, if you buy just vanilla WoW, you can experience the original world as it has been destroyed by the Cataclysm expansion. You can level up to level 60, but cannot access professions, areas, races, or other such things from any of the other expansions.
In order to access Outland, jewelcrafting profession, and Draenei? / Blood Elf races and starting areas, and level 61-70, you will need vanilla WoW + the Burning Crusade expansion.
In order to access Northrend, Inscription profession, and Death Knight class, and level 71-80, you will need vanilla Wow, Burning Crusade expansion, and Wrath of the Lich King expansion. Oh, and a prior level 55+ character on your account to play as a Death Knight.
In order to access the misc Cataclysm level 80+ areas, Archeology secondary profession, Worgen / Goblin races, and level 81-85, you will need vanilla WoW, Burning Crusade expanion, Wrath of the Lich King expansion, and the Cataclysm expansion.
Those are the restrictions. You can play on whatever realm you want with whoever you wish. You just may not be able to reach all the areas they can.
You can trial every bit of it with the 10 day trial, though I don't quite know how they treat it when your time is up.
You cannot play the original World of Warcraft as Cataclysm changed the map, starting areas, and other things significantly.
To clarify, this does not require you to buy any of the expansions, but it does require you to patch your game so that you're on the current version of the game. Even without buying the expansions there are many changes that will effect how you experience the game.
Best Answer
There are two different ways that a player could experience this type of rapid power increase.
Level Difference Scaling Mechanics
This is what you specifically encountered.
As you touched on there was a large "stat-squish" that took place with the launch of the Warlords expansion that made stat increases between expansions much more linear and much less exponential. One of the potential problems that this could have introduced was making older content harder even when significantly out levelling that content. So to prevent this issue they also introduced an algorithm that would both increase your damage output and also reduce damage taken when you are a higher level; this effect also grows as the level gap between you and the content increases.
In this scenario you can expect the "super powers" you experience to show up any time you are more than a level or two above what you are fighting.
Explosive ILvl Increase
At the end of each expansion there was extensive "End Game" content where you would keep getting better gear (same level requirement but higher iLvl). So at the time when the next expansion was released the baseline for the new gear would have an iLvl somewhere around the mid to high level range of the previous end game gear.
Now when you level a new character you are skipping all of that end game gear improvement so when you start into content for the next expansion you will see much higher iLvl increases when replacing pieces.
For example look at this quest from Ice Crown which offers iLvl 174 rewards and compare it to this quest from Mount Hyjal which offer iLvl 272 rewards.
Where someone who took part in the ICC raid would have had between iLvl 251(10 man normal) and 277 (25 man heroic).
Note: The power jumps come from how iLvl relates to the amount of stats on an item and is a topic too broad for this answer but the tl;dr; is higher iLvl = more stats per iLvl
In this scenario you can expect to see those "super powers" generally after you've earned a few new pieces from the new content and last up until the creature level catches up. Here are the level ranges for this:
*I'm pretty sure they re added the 2 level grace with legion