Learn Hardcore Russian: После семинара мы зайдём в кафе и обсудим проект. - After the seminar we will stop by a cafe and discuss the project.

Because после requires the genitive case. Семинар (nominative) changes to семинара (genitive singular).
Pattern: после + Genitive = “after (something)”.

Is После семинара a full sentence by itself, or what role does it play?

It’s a prepositional phrase functioning as an adverbial modifier of time (“after the seminar”). It sets the time frame for the main action: мы зайдём… и обсудим…

Why is мы included? Can it be omitted?

Yes, мы can be omitted if it’s clear from context, because Russian verb endings often show the person/number.

With subject: мы зайдём и обсудим (explicit: “we”)
Without: зайдём и обсудим (still “we”, but more context-dependent)

Including мы can add clarity or emphasis (“we, as opposed to someone else”).

What tense is зайдём and обсудим?