Die Glühbirne im Flur ist kaputt, also kauft meine Mutter morgen eine neue.

im is a contraction of in dem.

• der Flur = the hallway (masculine noun)
• With in for location, German often uses the dative case
• So in dem Flur becomes im Flur

So:

• der Flur → nominative
• in dem Flur → dative
• im Flur → the usual contracted form

You cannot say in Flur here because German normally needs an article in this kind of sentence.

Because Glühbirne is simply a feminine noun in German, so its dictionary form is:

• die Glühbirne = light bulb

This is something you usually just have to learn with the noun. Its gender affects other words in the sentence, such as:

• die Glühbirne (nominative)
• eine neue later in the sentence, where the missing noun is still understood to be feminine

kaputt means broken, not working, or out of order, depending on context.

In this sentence:

• Die Glühbirne ... ist kaputt = The light bulb ... is broken / doesn't work

For a light bulb, English might more naturally say has burnt out, but German often uses kaputt in everyday speech.

It comes after ist because it is acting like a predicate adjective:

• ist kaputt = is broken

This is because also counts as occupying the first position in the clause, so the conjugated verb must come next.

German main clauses usually follow the verb-second rule:

• Position 1: one element
• Position 2: the conjugated verb

So in:

• also kauft meine Mutter morgen eine neue

the structure is:

• also = position 1
• kauft = position 2
• meine Mutter = then comes the subject

This is a very common pattern in German.

Compare:

• Meine Mutter kauft morgen eine neue.
• Also kauft meine Mutter morgen eine neue.

Both are correct; the second just begins with also = so / therefore.

No, not usually.

In this sentence, also means:

• so
• therefore
• thus

So:

• Die Glühbirne im Flur ist kaputt, also kauft meine Mutter morgen eine neue.
• The light bulb in the hallway is broken, so my mother is buying a new one tomorrow.

This German also is a false friend for English speakers, because English also usually means too / as well.

German often leaves out a noun when it is obvious from context.

Here, eine neue means:

• a new one
• literally: a new [light bulb]

Because Glühbirne was already mentioned, repeating it is unnecessary.

This is very natural in German. The adjective neue takes the feminine ending because the omitted noun is feminine:

• die Glühbirne
• eine neue [Glühbirne]

The adjective ending depends on the article, gender, number, and case.

Here we have:

• eine = indefinite article
• implied noun = Glühbirne (feminine singular)
• the phrase is the direct object of kauft, so it is in the accusative
• feminine accusative with eine takes the adjective ending -e

So:

• eine neue Glühbirne
• shortened here to eine neue

That is why neue is correct.

German word order is flexible, but there are common patterns.

In the second clause:

• also kauft meine Mutter morgen eine neue

morgen is a time expression, and it is very common for time words to come before the direct object.

A very common order is:

• subject + verb + time + other elements

So this sounds natural:

• meine Mutter kauft morgen eine neue

You could move things around in some contexts, but this version is the most straightforward and natural.

meine Mutter is the subject of the second clause, so it is in the nominative case.

You can tell because she is the person doing the action:

• kauft = buys
• Who buys? meine Mutter

So:

• meine Mutter = nominative subject
• eine neue = accusative direct object

Not in this sentence.

• im Flur describes location: the light bulb is in the hallway
• den Flur would be accusative and would usually be used when there is movement into the hallway with in

Compare:

• Die Glühbirne ist im Flur. = The light bulb is in the hallway.
• Ich gehe in den Flur. = I go into the hallway.

Since the bulb is simply located there, im Flur is the correct choice.

Yes. kauft is present tense.

German often uses the present tense to talk about the near future when a time word makes the future meaning clear.

So:

• meine Mutter kauft morgen eine neue

literally looks like my mother buys tomorrow a new one, but naturally means:

• my mother is buying a new one tomorrow
• my mother will buy a new one tomorrow

This is completely normal in German.