Ohne Ausweis darf ich das Paket an der Paketstation nicht abholen.

German often puts a time/place/condition phrase first to set the scene. When Ohne Ausweis is placed in position 1, the verb still has to stay in position 2 (the V2 rule), so darf comes next and the subject ich moves after it:
Ohne Ausweis + darf + ich + …

Both are possible, but the word order changes depending on what comes first.

• Neutral: Ich darf das Paket … nicht abholen.
• With Ohne Ausweis in front: Ohne Ausweis darf ich … nicht abholen.
  This is just the normal consequence of V2 word order: the verb (darf) stays second.

Ohne normally takes the accusative. If you include an article, it would be accusative:

• Ohne einen Ausweis … (accusative masculine)

But it’s also very common to use ohne + noun with no article when speaking generally (like “without ID” as a general requirement):

• Ohne Ausweis … = without (any) ID / without having ID on you.

Ausweis is a general word for an ID or identification document. In everyday German it often means an official photo ID, typically:

• Personalausweis (German national ID card)
  But in context it can also include other acceptable IDs (passport, etc.), depending on the situation.

dürfen is about permission / being allowed to (rules, authorization).
können is about ability / possibility (can/could in the sense of capability).

So Ohne Ausweis darf ich … nicht … means “Without ID, I’m not allowed to …” (rule).
If you said Ohne Ausweis kann ich … nicht …, it would sound more like “Without ID, I can’t …” (practically impossible), which is close in meaning but less explicitly about permission.

Because abholen takes a direct object: you pick up something → accusative.
So: das Paket (accusative neuter).
Dative would be used for an indirect object (to/for someone), which you don’t have here.

• Paketstation is feminine → die Paketstation.
• an with a location (no movement, just “at”) uses dative → an der Paketstation.

So der here is dative feminine singular, not masculine nominative.

Often yes, but there’s a nuance:

• an der Paketstation = at the parcel station, right at/using it (very natural for machines/terminals)
• bei der Paketstation = near/by the parcel station (can sound a bit more like “in the vicinity of”)

For picking something up from the machine, an der Paketstation fits especially well.

nicht usually goes right before what it negates. Here the main action is abholen (in infinitive form at the end), so nicht goes immediately before that infinitive:
… das Paket … nicht abholen.

That means: the negation applies to the act of picking it up.

Because dürfen is a modal verb. In German, modal verbs take a second verb in the infinitive, and that infinitive goes to the end of the clause:
darf + (object/info) + abholen

Yes. abholen = ab- (separable prefix) + holen.

• In a simple present sentence without a modal: Ich hole das Paket … ab. (prefix goes to the end)
• With a modal verb, it stays together as an infinitive at the end: … darf ich … abholen.

It’s grammatical, but the emphasis changes and it can sound less natural.

• … das Paket … nicht abholen is the normal, neutral order.
• nicht das Paket tends to stress the package as the contrast, implying something like “I’m not allowed to pick up the package (but maybe something else).”