Cambridge has a decent explanation here. Aside from the missing commas, both of those sentences are possible.

Aside from the missing commas, both of those sentences are possible and correct, because 'after' makes the order of the events clear. The same is true of 'before'.

Clearer?

M-W gives the following as an example sentence for such a use of "after" (which avoids the inverted sentence structure in the OP):

: following the time when

I opened the door after she knocked.

Thus, the assumption in the OP that after + past perfect is the only correct form appears to be disproved. (Yep: English isn't German)

You could also say "After eating dinner, he went home".

In the first example, you're describing a series of events, all "equally" in the past, and not related: there's no connection between one event and the next. First he ate dinner, then he went home, then he wrote an e-mail and then he hung up a picture on the wall.

After he ate dinner, he went home, and after he wrote the e-mail, he hung up a picture.

In the second example you're making more of a connection between him eating dinner and going home. You imagine him waiting until he's finished his dinner before going home. He probably went home because he'd finished dinner and had nothing else planned.

"After he'd brushed his teeth, he went to bed" would sound more natural to me than "After he brushed his teeth, he went to bed", for example, as he was brushing his teeth in preparation for going to bed. There's a link.

The M-W example is also helpful.

After she'd knocked, he opened the door.

After she knocked, he opened the door.

--> Here, the second example sounds better to me, as knocking is so quick. It's not an activity you do for a while until it is completed. The past perfect expresses completion.

#5 (Yep: English isn't German)

Ha ha. So true in both directions :)

Zum weiteren Verständnis:

Ich hoffe jetzt, dass folgender Satz. wenn er denn überhaupt in Ordnung ist, das "had" benötigt. Liege ich da richtig?

Her DNA could still be traced on the door's surface in 1996, after she had knocked at the door in 1979.