Yes, your assumptions are restatements of rules except for the "before he gets in range" part; odds are it'll be easier to get into range and attack than to throw and then have to draw a new weapon. Both use two actions, but the first will use your Str bonus and be superior and the second requires you to begin combat wielding a weapon you're willing to throw away (so not your best weapon, a bad situation if you begin combat already in melee).
And a greataxe will have at best a 10' range increment, making it even worse, and it can't take a Returning Rune, so yeah, this is overall a poor choice, albeit cinematic. And throwing two is implausible.
Much of the rest will be up to GM adjudication being in improvised territory, but you can't expect it to remain 1d12 or be superior to other thrown weapons. You aren't throwing it as a proper axe, but with the same style (and I'd say about the same damage) as a rusty metal stove.
You'd do much better throwing an actual throwing weapon. Heck, even a simple javelin is better due to its range increment.
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If you're trying to milk "Rage at range w/ d12 weapon" I can't help you (nor will most GMs IMO). But if you want "savage Minotaur throwing and chopping", there's the Kineticist. Earth or Metal would suit chucking "axes" via Weapon Infusion. You could roleplay as barbaric as you'd like, and chuck elemental axes much further and better without worrying about Runes, drawing, GM variance, Dex-based attacks, or anything else.
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Yeah, this is a rather questionable tactic. Improvised weapon damage is up to the GM to decide on. And it is debatable if you can continue using the weapon runes that the weapon has when you use the weapon as an improvised weapon - the rules for runes don't say that they wouldn't still apply when the weapon is not wielded and used properly, but the improvised weapon rules don't say that they still use any fundamental or property runes that the weapon has when it is used as an improvised weapon. Neither side of the debate has any rule support for their stance on it.
So you may very well end up with a greataxe dealing 2d12 slashing damage when wielded and used properly, but when thrown as an improvised weapon the GM rules it to do 1d6 bludgeoning damage.
+1 for Metal Elemental Instinct Barbarian and getting some Impulse attacks with the Rage trait to use in place of an actual thrown greataxe.
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Using a weapon as an improvised weapon is an "off-label" use- the rules say improvised weapons are when you use something that isn't designed as a weapon as one. Weapons already have rules for how to wield them, and if you aren't wielding them that way, don't count on any of the other stats applying.
So, what can you do?
- Reflavor. Minotaurs are large, so you can take a large 1d6 throwing axe of some kind, and reflavor that as a greataxe for a smaller creature.
- Extending rune. It's a seventh level item, but for two actions you can strike with a 60ft. reach. Just reflavor it as a returning rune, and you get to throw your axe using strength for accuracy.
- As mentioned, use metal axes as a Kineticist blast.
The game does not support d12 thrown weapons, though, so it'd be pretty broken to play with what you're suggesting.
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Putting an extending rune on your greataxe and simply describing it as being thrown and returning to you might be your mechanically best option. At least then you get to retain your strength to attack and your high d12 damage. You also don't need extra feats and abilities to benefit from your rage bonus, as it's already included (though you can certainly keep Raging Thrower if you wish for other non-greataxe weapons).
The only real downside is it takes longer to "throw" (two actions instead of one).