Ykhjuy

As per the rule on PHB, page 195:

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.
If either weapon has the thrown property, you
can throw the weapon, instead of making a melee
attack with it.

Now, let's take for example a drow (MM, page 128):

Under its Actions, it can use a shortword, which is a light melee weapon. Should said drow find itself wielding a second shortsword, would it be able to use Two-Weapon Fighting within the normal limitations of the feature?

Yes, if you want it to be able to

NPCs in D&D 5e are designed differently from, and often do not follow the same rules as, the player characters - but in this case the same rules should apply. The guidance for reading monster stat blocks states:

When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures, such as the Dash or Hide action.

By a strict reading, the weapon attack options described in the monster's stat block are not uses of the Attack action - they're special actions that the monster can use, so if it uses one of those actions it's not taking the Attack action and so can't qualify for two-weapon fighting. (Functionally, these options are usually equivalent to taking the Attack action and making an attack with that weapon, and in most cases I'd personally rule as if they were the Attack action for situations where that's relevant, but that's not the rules as written.)

However, the monster explicitly retains the ability to use actions which are available to all creatures - including the Attack action - so it can take that action and make an attack with a weapon it's holding and then invoke two-weapon fighting just like a PC can.

Where things get a bit funky is when the Multiattack action is in play. A monster with Multiattack can make several attacks with a single action - which is much the same as a player character using the Attack action with the Extra Attack feature - but it is not the Attack action, so a monster that uses Multiattack cannot use those attacks to qualify for a two-weapon-fighting bonus action attack. A monster with Multiattack is probably better off using that than trying to use the two-weapon fighting rules though, since its Multiattack almost certainly has better stats and doesn't use up a bonus action.

Yes.

Two-weapon fighting rules apply to monsters per the MM's guidance on monsters and what actions they can take.

When a monster takes its action, it can choose from...one of the actions available to all creatures...as described in the Player's Handbook. (MM page 10.)

Further guidance on the next page of the MM specifically references the attack section of the PHB which contains the rules for Two Weapon Fighting.