You get two attacks with your main weapon, and one with your off hand weapon (if you want them).
Your attack action gets you two hits with your main weapon. After you take the attack action, you can then take a special bonus action to make the off hand attack. This is a single attack.
So yes, you get 3 attacks, two with your main weapon, and the third with your off hand weapon.
I'll get the simple bit done first - you're right, nothing in the Thrown property turns a melee weapon into a ranged weapon. It's a melee weapon you can use to make ranged attacks, so the Archery Fighting Style, which says that
You gain a +2 bonus to attack rolls you make with
ranged weapons.
can never apply to it. It can apply to darts, because they are thrown ranged weapons. This was also confirmed in a recent Sage Advice column:
Does the Archery fighting style work with a melee weapon that you throw? No, the Archery feature benefits ranged weapons. A melee weapon, such as a dagger or handaxe, is still a melee weapon when you make a ranged attack with it.
Now for the Dueling Fighting Style. Here, I have to disagree with you - when you roll damage for a thrown weapon, you aren't wielding that weapon in one hand. You were wielding it, then you threw it, then it hit. If you were still wielding it when it hit the target, it would be a melee attack. (Unless you threw it, then teleported across the battlefield and caught it right when it hit them, which sounds cool, but doesn't seem particularly useful.)
"Wielding" is a fairly nebulous term, so there is room for interpretation here, but personally I'm inclined to think that "wielding [...] in one hand" pretty clearly requires you to be holding something in your hand.
On the other hand, Crawford says that the Dueling Fighting Style does work with thrown melee weapons, and he's the authority on these matters.
Best Answer
Two-Weapon Fighting requires light, one-handed, melee weapons, which the stones are not
Nothing states that the stones are considered light or that they are melee weapons, in fact, they are used to make spell attacks not weapon attacks. Thus, they cannot be used with Two-Weapon Fighting. Even if you got the Dual Wielder feat they would still not work:
This still requires that you use one-handed melee weapons, which the stones are not (nor is the sling you could use with the stones).
It's unclear whether magic stone uses the Attack action
The spell might require its own unique action to hurl the stones and not the Attack action. I believe this to be the case as it makes a spell attack and does not explicitly mention the Attack action like other features do; for example, the Sun Soul Monk's Radiant Sun Bolt:
Because of this, the spell cannot benefit from Extra Attack as it requires the Attack action:
Either way, nothing is going to break
Any GM is perfectly allowed to rule either way and have the stones use the Attack action or not, though that would not change the fact that the stones are not light, melee weapons and cannot be used for TWF. I can't see any problems arising from allowing them to work with Extra Attack, so it would be a fine interpretation/ruling to me.
Personally, I would say magic stone requires its own action, but that's just me. I could easily see it being ruled or interpreted the other way around.