Learn German: Ein Junge steht auf dem Sprungbrett und sucht die Badehose. - A boy is standing on the diving board and is looking for the swimming trunks.

You are right that Junge is a weak noun (also called an n-declension noun), meaning it takes an -n ending in most cases. However, in the nominative case (when it is the subject of the sentence, like it is here), it stays Junge. You would only add the -n if it were in the accusative, dative, or genitive case. For example, as a direct object it becomes: Ich sehe den Jungen.

Why is it auf dem Sprungbrett (dative) rather than accusative?

auf is a two-way preposition. Because the verb stehen (to stand) describes a stationary position or location rather than movement towards a destination, we use the dative case (dem). If the sentence described him stepping onto the board (a change of location), you would use the accusative: Er tritt auf das Sprungbrett.

How could I use the Ge-...-e structure from this lesson to describe his endless searching?

You can take the verb stem such- (from suchen) and add the Ge- prefix and -e suffix to make the neuter noun das Gesuche. To express that his continuous searching is getting on your nerves, you could say: Sein ständiges Gesuche nach der Badehose nervt! (His constant searching for his swimming trunks is annoying!).

Why does the German sentence use instead of a possessive like ?