Q1. The right answer is A) ¿Dónde está la puerta?
In this case, you are asking for the door (puerta). So, given that the noun puerta is given in a singular form and this very noun matches the third-person singular, then it follows that you need to conjugate the verb estar as está.
Q2. The right answer is B. La silla está al lado del escritorio
In this case, la silla is the subject of the sentence, so since the corresponding pronoun matches the third-person singular (feminine), it follows that the conjugation of the verb estar is está.
Q3. The right answer is D. El ratón está detrás del teclado
In this case we have the subject el ratón. The explanation is the same as in the question Q2. The only difference is that the subject matches the third-person singular but in the masculine form.
Q4. The right answer is C. The flag is on the computer screen
bandera is a feminine noun, so it is true that the conjugation of the verb estar for this sentence is está.
Q5. The right answer is A. El estudiante está delante de la ventana
You can say this very sentence in other words as follows:
El estudiante está en frente de la ventana
Basically, the two sentences mean the same. So you are talking about a student who is in front of the window.
Q6. The right answer is B) My pencil sharpener is in my backpacks.
In this case, sharpener is a noun in singular form, so the conjugation of the verb estar is está for the subject matches the third-person singular.
Q7. The right answer is D) Los relojes están encima de la mesa
In this case, the subject los relojes is given in plural form. This subject matches the third-person plural. So the conjugation of the verb estar in this case is están.
Q8. The right answer is D) de las
La puerta está al lado de las sillas (The door is next to the chairs)
Sillas is a noun given in plural form (feminine). So the definite article that matches this noun is las.