Monitörü açıp parlaklığı düşürdüm.

It’s a converb meaning roughly “and (then) …”, linking two actions with the same subject in a tight sequence. The tense/person comes only from the main verb (düşürdüm), and applies to both actions.

• Forms: -ıp/-ip/-up/-üp (vowel harmony). After aç- you get açıp.
• It implies “first open, then lower.”
• Negative is possible: açmayıp = “without opening / not opening (and then …).”

Because it’s a definite direct object and must take the accusative. Turkish marks specific/known direct objects with -(y)ı/i/u/ü by vowel harmony.

• Monitör = “a/any monitor” (indefinite, no accusative)
• Monitörü = “the monitor” (definite accusative), which fits the context here.

If you wanted it indefinite, you’d say Bir monitör açtım (“I turned on a monitor”).

For the same reason: it’s a specific object, so it takes the definite accusative. Also note the consonant softening:

• parlaklık
  • -ı (accusative) → parlaklığı (k → ğ before a vowel)

Only if you explicitly mean “its brightness” (with a possessor). -sI is the 3rd person possessive, so:

• parlaklığı = “the brightness” (definite object, no possessor)
• monitörün parlaklığını = “the monitor’s brightness” (possessed + accusative)
• parlaklığını alone means “its brightness,” referring to a previously mentioned thing. In your sentence, without a stated possessor, parlaklığı is the neutral, context-driven choice.

• düş- (fall)
• -ür- (causative: “make fall, lower”) → düşür-
• -dü- (simple past of the “di-past,” harmonized to dü after a front rounded vowel)
• -m (1st person singular)

So düşürdüm = “I lowered.”

Both are fine, but -ıp is more compact and strongly suggests a quick, sequential link with the same subject. You could say:

• Monitörü açtım ve parlaklığı düşürdüm. (neutral coordination)
• Monitörü açıp parlaklığı düşürdüm. (tighter, more natural for sequential actions)

Yes. With -ıp, the action in the converb clause typically happens first. Swapping them changes the sequence:

• Monitörü açıp parlaklığı düşürdüm. = open → then lower
• Parlaklığı düşürüp monitörü açtım. = lower → then open (grammatical but odd in real life)

Only when the object is indefinite or generic. Compare:

• Su içtim. (“I drank water.” Indefinite, no accusative.)
• Suyu içtim. (“I drank the water.” Definite accusative.) In your sentence, parlaklığı is specific, so the accusative is expected. Parlaklık düşürdüm would sound off.

Yes, both are common:

• kısmak: very idiomatic for turning down brightness/volume. Parlaklığı kıstım.
• azaltmak: “to reduce/decrease,” neutral and formal. Parlaklığı azalttım. Düşürmek is also fine and common (“to lower”).

Yes. -ınca/ince means “when/once (I) …,” creating a temporal subordinate clause:

• açıp = “(I) opened and then …” (simple sequence)
• açınca = “when I opened …” (temporal frame; often implies causality/opportunity)