Learn Hardcore Swedish: Barnbarnet är nyfiket på festen men lite rädd för musiken. - The grandchild is curious about the party but a bit afraid of the music.

Here we have ett barnbarn → barnbarnet, so the correct form is nyfiket.
Spoken Swedish often drops the -t (you may hear barnbarnet är nyfiken), but in standard written Swedish nyfiket is the correct form.

Is barnbarn always a neuter word? Why?

Yes. Barnbarn is an ett-word: ett barnbarn, barnbarnet.
Many Swedish words ending in -barn are neuter, for example:

ett barn – a child
ett bonusbarn – a stepchild
ett barnbarn – a grandchild

So even though a grandchild is a person, the grammatical gender is neuter, and adjectives must follow that.

What’s the difference between barnbarnet, ett barnbarn, and barnbarnen?

ett barnbarn = a grandchild (indefinite, singular)
barnbarnet = the grandchild (definite, singular)
barnbarn (no article) can be plural in some contexts, but more clearly:
- barnbarn = grandchildren (indefinite plural, in context)
- barnbarnen = the grandchildren (definite plural)

In this sentence, barnbarnet means one specific grandchild: the grandchild.

Why is it nyfiket på festen? Why the preposition på?

With the adjective nyfiken / nyfiket (curious), Swedish normally uses på:

nyfiken på något – curious about something
- nyfiken på dig – curious about you
- nyfiken på boken – curious about the book

So nyfiket på festen literally is curious on the party, but the idiomatic English translation is curious about the party.
The preposition på here is mainly controlled by nyfiken, not by festen.

But på festen can also mean at the party, right? How do I know which it is here?

Yes, på festen on its own usually means at the party.
However, in this sentence, the structure is:

nyfiket på festen = curious about the party (because nyfiken på is a fixed pattern)

If you wanted to say at the party in this sentence, you would normally place på festen differently, for example:

Barnbarnet är på festen, nyfiket men lite rädd för musiken.
The grandchild is at the party, curious but a bit afraid of the music.

Why is it lite rädd and not liten rädd or litet rädd?

Here lite is an adverb meaning a little / a bit, not the adjective liten (small).

lite rädd = a bit afraid (degree adverb + adjective)
liten / litet / lilla = small / little (size)

So lite rädd describes how much the grandchild is afraid, not the size of the fear.

Why is it rädd för musiken and not rädd av musiken?

For feelings like fear, Swedish normally uses rädd för:

rädd för hundar – afraid of dogs
rädd för mörkret – afraid of the dark

So rädd för musiken = afraid of the music.
av is not used with rädd in this sense; rädd av musiken would sound wrong to native speakers.

Why is it musiken (definite) and not just musik?

Both are possible, but they mean slightly different things:

rädd för musik – afraid of music in general
rädd för musiken – afraid of the music (here: the specific music at this party)

In the sentence, musiken makes it clear we are talking about the particular music that is playing at the party.

Why doesn’t rädd change form to something like rädda with barnbarnet?

In Barnbarnet är lite rädd, rädd is a predicative adjective (after är, describing the subject).
Predicative adjectives agree in gender and number, but not in definiteness:

Huset är stort. (ett-word, neuter)
Bilen är röd. (en-word, common gender)
Bilarna är röda. (plural)

-a is used before a definite noun: den rädda flickan, det rädda barnet, de rädda barnen.
After är, we use the basic form (with -t for neuter if you wish to be formal), so barnbarnet är (lite) rädd is fine, and you don’t add a definite -a.

Can I change the order and say Barnbarnet är lite rädd för musiken men nyfiket på festen?

Yes, that word order is grammatically correct.
Both:

Barnbarnet är nyfiket på festen men lite rädd för musiken.
Barnbarnet är lite rädd för musiken men nyfiket på festen.

are fine.
The difference is mainly in emphasis: whichever description you place first tends to feel like the more prominent or expected characteristic.