Participial Adjectives

Add -d to the infinitive: schlafen gives schlafend, lachen gives lachend, kommen gives kommend, wachsen gives wachsend and brennen gives brennend. Before a noun the participle then takes an adjective ending, so schlafend plus -e becomes das schlafende Kind. It corresponds roughly to the English -ing adjective.

Partizip I shows an action still happening, Partizip II shows one already finished. Der schlafende Mann is sleeping right now, der eingeschlafene Mann has fallen asleep. Die kochende Suppe is on the boil, die gekochte Suppe is already cooked. Der ankommende Zug is arriving, der angekommene Zug has arrived.

Only as an adjective or adverb, not as a verb tense. German has no continuous tense, so I am sleeping is simply ich schlafe, never ich bin schlafend. But das schlafende Kind is perfectly normal, because there the participle modifies a noun and carries an adjective ending.

Yes, exactly the same endings as any other adjective. They agree with gender, case and number, and the ending also depends on whether a definite article, indefinite article or nothing precedes. Compare das gebrochene Fenster with ein gebrochenes Fenster and with die geschriebenen Briefe in the plural.

Almost any verb can form Partizip I, since you only add -d. Partizip II as an adjective works best with verbs that describe a resulting state, especially transitive ones: brechen gives gebrochen, schreiben gives geschrieben, kochen gives gekocht, öffnen gives geöffnet and verlieren gives verloren, as in die verlorene Zeit.