Woher kommt der Luftballon im Kinderzimmer? Gestern war er noch nicht da.

Woher means from where or where ... from, so it asks about origin.

• Wo kommt der Luftballon her? = Where does the balloon come from?
• Wo ist der Luftballon? = Where is the balloon?

In your sentence, the speaker is not asking where the balloon is located, but where it came from, so woher is the correct question word.

Because this is a question. In German, a wh-question usually has:

1. the question word
2. the verb
3. the subject

So:

• Woher = question word
• kommt = verb
• der Luftballon = subject

This is normal German word order for questions:

• Woher kommt der Luftballon ...?
• Wann kommt er?
• Warum ist er da?

No. Der Luftballon is the subject of the sentence, so it stays in the nominative case.

The verb kommen does not take a direct object here. The balloon is the thing doing the coming, so it is the subject:

• der Luftballon kommt

That is why it is der, not den.

Im is a contraction of in dem.

So:

• im Kinderzimmer = in the children's room / in the child's bedroom

German often combines a preposition and article like this:

• in dem → im
• an dem → am
• zu dem → zum
• bei dem → beim

So im Kinderzimmer is just the natural shortened form.

Because im Kinderzimmer describes a location, not movement into the room.

A useful German distinction is:

• wo? = location → usually dative
• wohin? = direction/movement toward a place → usually accusative

Compare:

• Der Luftballon ist im Kinderzimmer. = He is in the room.
• Der Luftballon fliegt ins Kinderzimmer. = He flies into the room.

In your sentence, the room is the place where the balloon is, so im Kinderzimmer is correct.

Because German very often forms compound nouns.

• Kinder = children
• Zimmer = room

Together:

• Kinderzimmer = children's room / nursery / child's room

This is extremely common in German:

• Haus + Tür → Haustür
• Schlaf + Zimmer → Schlafzimmer
• Luft + Ballon → Luftballon

The noun is:

• der Luftballon = masculine

That matters because it affects articles and pronouns:

• der Luftballon = nominative
• den Luftballon = accusative
• dem Luftballon = dative

And later in the second sentence, it is referred to with er because Luftballon is masculine:

• Gestern war er noch nicht da.

Because er refers back to der Luftballon.

In German, pronouns match the grammatical gender of the noun, not necessarily natural gender.

Since Luftballon is masculine (der Luftballon), the pronoun is:

• er = he / it

Here, in English we would normally say it, but German uses er because the noun is masculine.

Noch nicht means not yet.

Da often means there or here/present, depending on context. In this sentence:

• Gestern war er noch nicht da.
• literally: Yesterday he was not yet there/present.
• natural meaning: Yesterday it wasn't there yet.

So da sein is a very common expression meaning to be there / to be present / to be around.

German main clauses usually follow the verb-second rule: the finite verb is in the second position.

If Gestern comes first, then the verb must come next:

• Gestern war er noch nicht da.

This is correct because:

1. Gestern = first element
2. war = second element
3. er = after the verb

You could also say:

• Er war gestern noch nicht da.

That is also grammatical, but it gives a slightly different emphasis. Starting with Gestern highlights the time.

Because the two sentences talk about different times and use different verbs.

• kommt = present tense of kommen
• war = simple past of sein

So:

• Woher kommt der Luftballon im Kinderzimmer?
  The speaker is asking now about the balloon's origin.

• Gestern war er noch nicht da.
  This refers to yesterday, so past tense is used.

Also, in everyday German, war is very common for talking about the past with sein.

No, that would be wrong here.

Woher already contains the idea of from where, so you do not add another from phrase unless you really mean the balloon is coming out of the room.

Compare:

• Woher kommt der Luftballon im Kinderzimmer?
  = Where does the balloon in the children's room come from?

• Woher kommt der Luftballon?
  = Where does the balloon come from?

• Kommt der Luftballon aus dem Kinderzimmer?
  = Is the balloon coming out of the children's room?

So im Kinderzimmer describes which balloon you mean: the one located in the children's room.

In this sentence, it most naturally identifies which balloon is being talked about:

• the balloon in the children's room

So it functions like a description of der Luftballon.

The sentence is essentially:

• Woher kommt [der Luftballon im Kinderzimmer]?

This is different from a sentence where the location belongs more strongly to the action:

• Woher kommt der Luftballon ins Kinderzimmer? would suggest movement into the room, but that is not what your sentence says.

German and English build questions differently.

English often keeps the subject before the verb:

• Where does the balloon come from?

German puts the finite verb directly after the question word:

• Woher kommt der Luftballon?

Also, German can express where ... from with the single word woher, which English cannot do in the same way. So the structure is different, even though the meaning matches.

Yes, Luftballon is a very common and normal word for a toy balloon or party balloon.

Literally it is:

• Luft = air
• Ballon = balloon

So Luftballon is the usual everyday word for the kind of balloon you would find in a child's room.