Learn Hardcore Russian: Я настроил яркость монитора на минимум, чтобы глаза меньше уставали. - I set the monitor brightness to minimum so my eyes would get less tired.

Настроил is a perfective past form of настроить. Perfective here means the action is viewed as completed: you adjusted the brightness (and finished doing it).
If you used imperfective настраивал, it would suggest a process/habit/background action: you were adjusting it (maybe repeatedly, or over some time), without focusing on the completed result.

Why is it Я настроил (masculine), and what changes for a female speaker?

Russian past tense agrees in gender and number.

Male speaker: Я настроил
Female speaker: Я настроила
Plural (we): Мы настроили

Why is яркость in this form, and what case is it?

Яркость is the direct object of the verb настроил, so it’s in the accusative case. For this feminine noun, accusative singular looks the same as nominative singular: яркость.

Why is it яркость монитора and not something like яркость монитор?

Монитора is genitive singular of монитор. The genitive is used to show possession/association: brightness of the monitor.
So яркость монитора literally means the monitor’s brightness / the brightness of the monitor.

What does на минимум mean grammatically, and why is it на + this form?

На минимум means setting something to a target value, like moving a slider to a point.
With на, Russian uses:

accusative for movement/change to a new value: на минимум (to minimum)
prepositional for being at a location/state: на минимуме (at the minimum)

So на минимум emphasizes changing the setting to that value.

Is на минимум the only natural way to say this? What are alternatives?

It’s very common, but alternatives include:

Я снизил яркость монитора до минимума (I reduced it down to the minimum)
Я поставил яркость на минимум (more colloquial: set it to minimum)
Я выставил яркость на минимум (also common, especially for device settings)

All are natural; they differ slightly in style and emphasis.

Why is чтобы used, and what kind of clause does it introduce?

Чтобы introduces a purpose clause: the goal/intention behind the action.
Structure: main action + чтобы + result you want.
Here: you adjusted the brightness so that your eyes would get tired less.

Why is it глаза (nominative) and not some other case?

In the чтобы clause, глаза is the grammatical subject of уставали (were getting tired). Subjects are normally nominative, so глаза stays nominative plural.

Why is the verb уставали in past tense, when the meaning is about what happens after the adjustment?

In Russian, purpose clauses with чтобы often match the time frame of the main clause, especially in past narration. So you get a past form: чтобы глаза меньше уставали.
English uses so that my eyes get tired less / wouldn’t get so tired, but Russian commonly uses past imperfective here even though it refers to the intended effect after the action.

Why is it уставали (imperfective) and not устали (perfective)?

уставали (imperfective) = getting tired as a general/ongoing tendency (less fatigue over time)
устали (perfective) = became tired (a completed event, like at one specific moment)

The sentence is about reducing ongoing eye strain, so imperfective уставали fits best.

What does меньше do here, and what part of speech is it?

Меньше is a comparative word meaning less. Here it modifies the verb уставали (got tired), so it functions like an adverb: got tired less / were less tired.

Is the word order flexible here? Could I move parts around?

Yes, Russian word order is flexible, though some orders sound more neutral than others. For example:

Я настроил яркость монитора на минимум, чтобы глаза меньше уставали. (neutral)
Чтобы глаза меньше уставали, я настроил яркость монитора на минимум. (emphasizes the purpose first)
Я настроил на минимум яркость монитора... (possible, a bit more emphasis on the target setting)

The meaning stays basically the same; the focus shifts.