[RPG] Unarmed Strikes/Martial Arts with Potent Psionics

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 have to have 14 cleric levels. Class features require levels in the class, and don't benefit from levels in other classes except when otherwise stated. The reason that your spellcasting improves when you multiclass with fighter is that the rules for multiclassing are one such exception, explicitly counting one-third of your fighter levels (assuming you're an Eldritch Knight figther, of course, and rounded down) toward the Multiclass Spellcasting feature (p. 164).

To be doubly sure, the PHB restates at the beginning of the Multiclassing section on Class Features (ibid) that you gain class features from class levels, with only a few explicit exceptions to that rule:

When you gain a new level in a class, you get its features for that level. A few features, however, have additional rules when you’re multiclassing…

Another notable exception to this rule is the improved versions of cantrips, which explicitly work off your total levels rather than your spellcasting levels.