Mein Vater holt die Leiter, um die Lampe im Flur zu wechseln.

Holen usually means to fetch, to go and get, or sometimes to bring back.

So Mein Vater holt die Leiter suggests that the father is going to get the ladder, not that he already has it in his hands.

That is different from:

• nimmt die Leiter = takes the ladder
• trägt die Leiter = carries the ladder

So holt is a very natural choice if someone goes to get the ladder because they need it for something.

Die Leiter is the direct object of holt, so it is in the accusative case.

The noun Leiter is feminine:

• nominative: die Leiter
• accusative: die Leiter
• dative: der Leiter
• genitive: der Leiter

In this sentence, the form happens to look the same in nominative and accusative because many feminine nouns do that.

Because both nouns are feminine singular:

• die Leiter = the ladder
• die Lampe = the lamp

Also, in the accusative singular, feminine nouns still use die, so:

• die Leiter after holt
• die Lampe after wechseln

That is why both articles are die, even though they have different jobs in the sentence.

Um ... zu + infinitive is a very common structure used to express purpose:
in order to ...

So:

• um die Lampe im Flur zu wechseln = in order to change/replace the lamp in the hallway

This tells you why the father is getting the ladder.

A useful pattern is:

• ..., um etwas zu tun = ..., in order to do something

Example:

• Ich lerne Deutsch, um in Deutschland zu arbeiten.
• I’m learning German in order to work in Germany.

Because in an um ... zu clause, the infinitive normally goes to the end.

So the structure is:

• um
  • other elements + zu
    • infinitive at the end

Here:

• um die Lampe im Flur zu wechseln

This final position is very typical in German subordinate or infinitive constructions.

Because the part starting with um ... zu is an infinitive clause, and in German these clauses are normally separated by a comma.

So:

• Mein Vater holt die Leiter, um die Lampe im Flur zu wechseln.

That comma is standard and important in written German.

Because after um you use an infinitive with zu, not a conjugated verb.

Compare:

• Er wechselt die Lampe.
  Here wechselt is a normal conjugated verb.

• ..., um die Lampe zu wechseln.
  Here wechseln is an infinitive, so it becomes zu wechseln.

So wechselt would not fit after um.

Yes, the normal understanding is that the same subject from the main clause is also the understood subject of the um ... zu clause.

So in:

• Mein Vater holt die Leiter, um die Lampe im Flur zu wechseln.

the natural meaning is:

• My father gets the ladder in order to change the lamp in the hallway.

If German wants to show that someone else is doing the second action, it usually uses a different structure, not um ... zu.

Im is a contraction of in dem, so it is dative:

• im Flur = in the hallway

Here it describes location: the lamp is in the hallway.

German often uses:

• dative for location: where?
• accusative for direction/movement toward somewhere: where to?

So:

• die Lampe im Flur = the lamp in the hallway
• in den Flur gehen = to go into the hallway

Since the sentence is talking about where the lamp is, im Flur is correct.

In everyday German, eine Lampe wechseln often means to replace a lamp or, depending on context, to change the bulb/light.

In real life, speakers may use Lampe a bit loosely. Sometimes they mean:

• the whole lamp
• the light bulb
• the light fixture, depending on context

So the exact physical object may depend on the situation, but grammatically wechseln here means to replace/change.

Because mein Vater is the subject of the sentence, so it is in the nominative case.

The verb holt agrees with mein Vater:

• Mein Vater holt ...

If father were a direct object, you would get a different form:

• Ich sehe meinen Vater.

So:

• nominative: mein Vater
• accusative: meinen Vater

Yes, die Leiter meaning ladder is feminine.

Be careful, though: German has some words that look similar but mean different things. For example:

• der Leiter = the leader/head of something
• die Leiter = the ladder

So the article matters a lot here. In your sentence, die Leiter clearly means the ladder.

Yes. German has other ways to express purpose, but um ... zu is one of the most natural when the subject is the same.

For example, you could say something like:

• Mein Vater holt die Leiter, weil er die Lampe im Flur wechseln will.

That means:

• My father gets the ladder because he wants to change the lamp in the hallway.

But um ... zu is shorter and very common when you mean in order to.