Learn Farsi: من گواهی را گرفته، به مرز رفتم. - Having gotten the certificate, I went to the border.

This is the clause chaining structure you are learning in this lesson! Instead of using a conjunction like "and" with two fully conjugated verbs (گرفتم و رفتم), the first action is turned into a past participle (گرفته). Only the final verb in the sentence (رفتم) takes a personal ending to show who did the action. It translates smoothly to "Having gotten...".

Could I use this structure if someone else got the certificate, but I went to the border?

No. You can only link verbs with a past participle if they share the exact same subject. Because the final verb رفتم means I went, the participle گرفته must also mean that I got the certificate. If the subjects are different, you must use separate conjugated clauses.

Is this "having done X..." structure common in everyday conversation?

Not really. Linking actions with a past participle is quite formal and is mostly used in written Farsi, literature, or news broadcasts. In everyday spoken Farsi, people usually just conjugate both verbs and link them with "and" (و), sounding more like: من گواهی رو گرفتم و رفتم مرز.

What kind of "certificate" does گواهی refer to?

گواهی is a general word for a certificate, credential, or piece of written proof. Depending on the context, it could be a medical clearance, an educational certificate, or a legal permit that you might need to show the guards at the border.