The definite article would be required here, even if the meaning of the sentence was general (if the speaker was talking about dropping out of school in general, not just going home and leaving the building). Of course, we can't know that without specific context.
It could have been omitted in other cases, like "Πηγαίνω σχολείο". While the article could be used, it's omitted, because it's a general statement. Just like πηγαίνω στρατό, πηγαίνω σπίτι, πηγαίνω Αθήνα would make sense as well.
It's not clear from the exercises that English present-perfect and Greek present-perfect have the same uses (there simply isn't enough context). Can you give me some contrasting examples where as a Greek speaker you would select between the aorist and the periphrastic present perfect?
I do believe that the English Present Perfect and the Greek Present Perfect have the same uses. According to almost every Greek grammar book, Present Perfect (Παρακείμενος) is used to describe an action in the past (that's usually completed) with consequences in the present time. I do believe that this is the use of the Engish Present Perfect as well.
If the Greek sentence is in Present Perfect, the translation has to match the tense, for learning's sake. Even if the context would be almost the same with Past Simple (Αόριστος), we have to set a limit.
The difference becomes clear in sentences like
I drank wine last night - Ήπια κρασί χθες το βράδυ.
(Έχω πιει κρασί χθες το βράδυ would be wrong. The action has no consequences in the present, it's just an action that was completed in the past.)
Έχω ήδη φάει, δεν πεινάω - I have already eaten, I'm not hungry.
(Ιn this case, the action does have consequences in the present, even though the action of was completed in the near past. The speaker is not hungry now.)
It's all about the context and words that indicate which tense should be used (last night, yet, already etc.). If there is no context or key word, the tenses of the translation and sentence have to match. ^.^