Da die Glühbirne im Flur kaputt ist, lasse ich die Jalousie am Morgen länger offen.

No. Here da means since / because, not there.

German da can be:

• an adverb meaning there
• a subordinating conjunction meaning since / because

In this sentence, it introduces the reason:

Da die Glühbirne im Flur kaputt ist, ...
= Since/Because the light bulb in the hallway is broken, ...

Because da introduces a subordinate clause.

In German subordinate clauses, the finite verb normally goes to the end:

• die Glühbirne ist kaputt → main clause
• da die Glühbirne kaputt ist → subordinate clause

So ist moves to the end because of the conjunction da.

German main clauses follow the verb-second rule.

When the sentence starts with a subordinate clause, that whole clause takes the first position. Then the finite verb of the main clause must come immediately after it:

• Da die Glühbirne im Flur kaputt ist, = position 1
• lasse = position 2
• ich = then the subject follows

So:

• Da ..., lasse ich ... ✔
• Da ..., ich lasse ... ✘

Both can mean because.

In many cases, both are possible:

• Da die Glühbirne im Flur kaputt ist, ...
• Weil die Glühbirne im Flur kaputt ist, ...

A rough difference:

• weil is the most common everyday word for because
• da often sounds a bit more formal, written, or like the reason is already known/background information

Also, da is especially common when the reason comes first in the sentence.

Because German uses commas to separate a subordinate clause from a main clause.

So in:

Da die Glühbirne im Flur kaputt ist, lasse ich ...

the comma is required.

If the order is reversed, you would also use a comma:

Ich lasse die Jalousie am Morgen länger offen, da die Glühbirne im Flur kaputt ist.

Because kaputt is being used as a predicate adjective, not directly before a noun.

Compare:

• Die Glühbirne ist kaputt. → predicate adjective, no ending
• die kaputte Glühbirne → adjective before a noun, with ending

So after verbs like sein, werden, and bleiben, adjectives normally do not take endings.

im is a contraction of in dem.

So:

• im Flur = in dem Flur

Here it describes a location, not movement, so German uses the dative case after in.

That is why it is im Flur, not in den Flur.

Compare:

• Die Glühbirne ist im Flur. → location, dative
• Ich gehe in den Flur. → movement into, accusative

It is nominative, because it is the subject of the verb ist.

You can think of it as the thing that is broken:

• die Glühbirne = the light bulb
• ist kaputt = is broken

So die Glühbirne is the subject of the subordinate clause.

Here lassen means to leave something in a certain state.

So:

• Ich lasse die Jalousie offen. = I leave the blind open.

Structure:

• lasse = verb
• die Jalousie = direct object
• offen = describes the state the object is left in

This is a very common pattern:

• Ich lasse das Fenster offen.
• Sie lässt die Tür zu.

So in this sentence, it does not mean allow. It means leave.

Jalousie is a feminine noun: die Jalousie.

Here it is the direct object of lasse, so it is in the accusative case. But for feminine singular, nominative and accusative look the same:

• nominative: die Jalousie
• accusative: die Jalousie

So the form does not change, even though the case is accusative.

länger means longer and modifies offen.

So:

• offen = open
• länger offen = open for longer / open longer

This is the natural German order here. The comparative word länger comes before the adjective/adverb it modifies.

Compare:

• Ich lasse das Fenster länger offen.
• Sie bleibt heute länger wach.

So länger offen works like longer open in structure, even though idiomatic English often says open longer.

am is a contraction of an dem.

In time expressions, am Morgen means in the morning.

So:

• am Morgen = in the morning
• am Abend = in the evening
• am Montag = on Monday

This is a fixed and very common time phrase in German.

Yes:

Ich lasse die Jalousie am Morgen länger offen, da die Glühbirne im Flur kaputt ist.

That is also correct.

The difference is mostly about focus:

• starting with Da ... puts the reason first
• starting with Ich lasse ... puts the action first

Both are natural German.

Not exactly.

• die Glühbirne = the light bulb
• die Lampe = the lamp or sometimes the whole light fixture

So if the actual bulb is broken, Glühbirne is the precise word.

In everyday speech, people are not always perfectly precise, but in this sentence Glühbirne specifically refers to the bulb.