[RPG] Monk Proficiency with Spears/Javelins

No. You can't dual wield Unarmed Strikes because they are not considered a Light weapon. That said, I don't believe it would be gamebreaking to houserule it that way.

However, in any case, you can't use two-weapon fighting and the Monk's martial arts on the same turn, because each uses a bonus action, and you only get one bonus action per turn. So your example of "tri-wielding" with a quarterstaff attack, an off-hand Unarmed Strike, and then a Martial Arts bonus Unarmed Strike would not be possible, because it would require two bonus actions to pull off.

Two weapon fighting:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in your other hand.

Martial Arts:

When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.

Note that the Monk Martial Arts ability does allow the Monk to effectively dual wield Unarmed Strikes, since they can unarmed strike as their main Attack action, and as their bonus action. And unlike regular Two Weapon Fighting, they get the bonus to their damage roll on the bonus attack as well.

You can wield & attack with two weapons without any special class feature or feat.

Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):

Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.

Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.

The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.

Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.