Express uncertainty mood, using modal verbs

Question

I failed to understand the different meanings between those sentences with the modal verb "dürfen" in regard to expressing Uncertainty in German language. It looks like this is a very complicated issue as there is no absolute Comprehensive Guide on this in the Internet. (feel free to point me to the right websites/books)

1. Er darf den Apfel essen.
2. Er darf den Apfel gegessen haben.
3. Er dürfte den Apfel essen.
4. Er dürfte den Apfel gegessen haben.
5. Er durfte den Apfel essen.
6. Er durfte den Apfel gegessen haben.

The 1) is easy: "He is allowed to eat the apple." But the rest are quite vague. And I'm not quite sure some of them are even correct! The 3) and 4) use subjunctive tense.

We can construct the same structure with other modal verbs, e.g. "können", "müssen". E.g. "Er müsste den Apfel gegessen haben". But I got the same issue: how different is it between "Er müsste den Apfel gegessen haben" and "Er muss den Apfel gegessen haben"?

Answer 1

Er darf den Apfel essen.

He is allowed to eat the apple.

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Er darf den Apfel gegessen haben.

This usually makes no sense as essen happens before dürfen. You can use it in some corner cases when you express a wish about an uncertain even in the past with dürfen. But then it's usually nicht dürfen:

Sie darf den Apfel nicht gegessen haben.

She must not have eaten the apple. Poor Snow White!

Outside of such a corner case, flip modal and perfect auxiliary:

Er hat den Apfel essen dürfen.

He was allowed to eat the apple.

With other modals as können for example both variants make sense.

Er kann den Apfel gegessen haben.

It is possible that he ate the apple.

Er hat den Apfel essen können.

He was able to eat the apple.

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Er dürfte den Apfel essen.

He likely eats the apple.

or

He is allowed to eat the apple if he wanted.

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Er dürfte den Apfel gegessen haben.

He has likely eaten the apple.

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Er durfte den Apfel essen.

He was allowed to eat the apple.

This is the same as Perfekt. Northern speakers use Präteritum in place of Perfekt for the auxiliaries, the modals, and some very common verbs.

Also, within a story, that "durfte" is the current event within the story. Same as narration in English is also told with simple past tense.

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Er durfte den Apfel gegessen haben.

This is ungrammatical. You can't combine Präteritum with Perfekt. If you want to make it Plusquamperfekt, that has to be phrased as:

Er hatte den Apfel essen dürfen.

He had been allowed to eat the apple.