Jeg opdager, at min billet stadig ligger i lommen.

Because standard Danish spelling normally uses a comma before a subordinate clause introduced by at.

So in:

Jeg opdager, at min billet stadig ligger i lommen.

the part after at is a subordinate clause: at min billet stadig ligger i lommen.

In English, you often would not need a comma there, but in Danish it is very common and standard.

At means that and introduces a subordinate clause.

So:

• Jeg opdager = I discover / I realize
• at min billet stadig ligger i lommen = that my ticket is still in the pocket

It connects the main idea to the information being discovered.

In Danish, a possessive word like min, din, hans, vores, etc. normally replaces the article.

So you say:

• min billet = my ticket
• not min billetten

This is similar to English: you say my ticket, not my the ticket.

Danish often uses the definite form for body parts, clothing, and things closely connected to a person where English prefers a possessive.

So:

• i lommen literally = in the pocket
• but in context it often means in my pocket

This is very natural in Danish. You can say i min lomme, but i lommen often sounds more idiomatic when it is obvious whose pocket is meant.

Because this is a subordinate clause after at, and Danish word order changes in subordinate clauses.

In a main clause, you often get:

• Min billet ligger stadig i lommen.

But after at, the adverb usually comes before the finite verb:

• at min billet stadig ligger i lommen

So this is a very useful pattern:

• main clause: subject + verb + adverb
• subordinate clause: subject + adverb + verb

Stadig means still.

It shows that the situation continues to be true. The ticket was in the pocket before, and it remains there now.

Examples:

• Den er stadig her. = It is still here.
• Jeg venter stadig. = I am still waiting.

Danish often uses position verbs like:

• ligge = lie
• stå = stand
• sidde = sit

instead of a general be.

For objects in a place, Danish often prefers one of these more specific verbs. A ticket in a pocket is naturally described as ligger.

So:

• Billet ligger i lommen sounds idiomatic.
• Billet er i lommen is understandable, but less natural in many contexts.

Not necessarily in a strict physical sense.

Yes, ligge originally relates to lying, but in everyday Danish it is often just the normal verb for where an object is located. So here it does not force you to imagine the ticket carefully lying flat in the pocket. It simply sounds like the natural way to describe its location.

Either can work, depending on context.

Opdage often means discover, but in sentences like this it can also feel close to realize or notice:

• I discover that my ticket is still in my pocket
• I realize that my ticket is still in my pocket
• I notice that my ticket is still in my pocket

If the meaning has already been shown to the learner, the main thing to remember is that opdager introduces the moment of becoming aware of something.

Yes. Opdager is the present tense of opdage.

So:

• jeg opdager = I discover / I am discovering / I realize
• jeg opdagede = I discovered / I realized

As in English, the simple present can sometimes describe an action happening at the moment of narration, especially in examples or storytelling.

Yes. Stadigvæk is possible and means roughly the same thing as stadig, often with a slightly more conversational or emphatic feel.

So you could say:

• Jeg opdager, at min billet stadigvæk ligger i lommen.

But stadig is shorter, very common, and perfectly natural here.

Yes, very much so. It combines several common Danish features:

• a main clause: Jeg opdager
• a subordinate clause introduced by at
• subordinate-clause word order: min billet stadig ligger
• a possessive noun phrase: min billet
• a position verb: ligger
• the definite form after a preposition: i lommen

So it is a great example of how natural Danish often differs from direct English word-for-word structure.