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 most likely will not be able to use Martial Arts as an animal, but not for the reason you brought up.
There are two factors to consider: the limitations of the Martial Arts feature, and those of the Wild Shape feature.
Part of the description of the monk's Martial Arts feature reads (PHB, p. 78):
You gain the following benefits while you are unarmed or wielding only
monk weapons and you aren't wearing armor or wielding a shield:
Armor and shield are obvious, most animals will not use them and natural armor is not "worn". So can you not wield your natural weapons? In an unofficial tweet in December 2016, rules designer Jeremy Crawford said that:
No general rule gives "hold" or "wield" meaning beyond the English
(the words and their meaning in a particular context).
The OED's definition of "wield" states:
Hold and use (a weapon or tool)
So if you simply do not use something, you are no longer wielding it. You can certainly do that.
The description of the druid's Wild Shape feature says, in part (PHB, p. 67; emphasis mine):
You retain the benefit of any features from your class, race, or other
source and can use them if the new form is physically capable of doing
so.
Can a bear perform martial arts maneuvers you have learned as a human? Can a crocodile? This will be up to a DM ruling, but the answer will most likely be no. The only case where I personally would rule yes is if you take the shape of an ape or monkey.
Best Answer
No.
armed in this sense means wielding a weapon. The wording of the rule implies ("unarmed or wielding ... ") that characters not wielding weapons are unarmed.
Just because an object the monk happens to be holding could be used as an improvised weapon doesn't mean they are wielding a weapon. If the monk was swinging a lit torch like a club, they would be wielding it as an improvised weapon (and wouldn't be unarmed) ... except most improvised weapons are clubs, and clubs are monk weapons.
As another example, if a monk picked up a fallen character's battleaxe so they could give it to an ally, they aren't necessarily wielding it, and so they don't necessarily stop being unarmed.