Learn Hardcore Indonesian: Kucing saya bersembunyi di balik sofa.

The verb bersembunyi already functions as is hiding, so you don’t need anything like is.
You only use adalah (similar to is/are) mainly before nouns, not before normal verbs:

Kucing saya adalah hewan. = My cat is an animal.
But: Kucing saya bersembunyi. (not Kucing saya adalah bersembunyi)

Why is it Kucing saya and not Saya kucing for my cat?

In Indonesian, the possessed noun comes first, and the possessor (the owner) comes after:

kucing saya = my cat (literally: cat I)
buku kamu = your book
rumah mereka = their house

Saya kucing would mean something like I (am) a cat, which is not what you want here.
So Kucing saya bersembunyi… = My cat is hiding…

Can I also say kucingku instead of kucing saya? Is there a difference?

Yes, you can.

kucing saya = my cat (neutral, slightly more formal/polite)
kucingku = my cat (more informal / personal / often used in speech, writing, and songs)

Both are correct. The difference is mostly in tone, not in grammar:

Kucing saya bersembunyi di balik sofa.
Kucingku bersembunyi di balik sofa.

Both mean My cat is hiding behind the sofa.

What does bersembunyi mean, and what is the role of ber-?

The root word is sembunyi = to hide / hidden.
The prefix ber- often turns roots into intransitive verbs (actions without a direct object).

bersembunyi = to be in the state of hiding / to hide oneself

So Kucing saya bersembunyi = My cat is hiding.

In casual speech, people sometimes drop ber- and say Kucing saya sembunyi di balik sofa, but bersembunyi is the more standard/complete form.

(If you want to hide something (an object), you’d use menyembunyikan, a different verb.)

Why is the preposition di used here? What does it mean?

Di is a preposition meaning in / at / on / inside / at the location of.
It marks a location, not movement.

di rumah = at home
di sekolah = at school
di balik sofa = behind the sofa (at the place behind the sofa)

If you want to express movement towards a place, you usually use ke (= to/towards):

Kucing saya lari ke belakang sofa. = My cat ran to behind the sofa.

What is the difference between di balik and di belakang? Both seem to mean behind.

Both can be translated as behind, but there is a nuance:

di belakang = behind, at the back of (more general, spatial back/behind)
di balik = behind / on the far or hidden side of something, often suggesting hidden or on the other side

In many contexts they are interchangeable:

Kucing saya di belakang sofa.
Kucing saya di balik sofa.

Both can mean My cat is behind the sofa, but di balik sofa fits especially well with bersembunyi, because it emphasizes the idea of being hidden from view.

Can balik stand alone? What does it mean on its own?

Yes. Balik has several related meanings, depending on context:

As part of di balik:
- di balik pintu = behind the door / on the other side of the door
As a verb root balik: to return / to go back / to turn over
- Saya balik ke rumah. = I go back home.
- Balikkan buku itu. = Turn that book over.

In di balik sofa, it works with di to mean on the hidden/other side of the sofa.

Why is there no word for the or a before kucing? How do you say the cat or a cat?

Indonesian does not use articles like a/an/the.
Kucing alone can be a cat or the cat, depending on context.

Kucing saya can mean my cat (and you already know which one).
To make it clearly that (specific) cat, you can add itu (that):
- Kucing saya itu bersembunyi di balik sofa.
  = That cat of mine is hiding behind the sofa.

There is no separate word for a/an; indefiniteness is usually inferred from context or added with words like seekor (a classifier for animals), e.g. seekor kucing = a cat (one cat).

How do I show tense? How would I say was hiding or will hide?

Indonesian verbs do not change form for tense.
You add time words or particles to show when something happens.

Kucing saya bersembunyi di balik sofa.
→ My cat hides / is hiding / was hiding (tense depends on context).

To be explicit:

tadi / tadi malam = earlier / last night
- Tadi malam kucing saya bersembunyi di balik sofa.
  = Last night my cat was hiding behind the sofa.
nanti / besok = later / tomorrow
- Besok kucing saya mungkin bersembunyi di balik sofa.
  = Tomorrow my cat might hide behind the sofa.

Aspect (continuous vs simple) is also usually left to context; bersembunyi itself can be hides or is hiding.

Is di written separately or together with the next word? I’ve seen di- as a prefix too.

Good observation.

di as a preposition of place (in/at/on) is written separately:
- di rumah, di sekolah, di balik sofa
di- as a passive verb prefix is written together with the verb:
- dibaca = is read / was read
- ditulis = is written

In di balik sofa, di is a preposition, so it must be separate: di balik, not dibalik (that would be a verb form meaning turned over / flipped).

Is sofa an Indonesian word? Are there other ways to say sofa or couch?

Sofa is a loanword but is completely standard in modern Indonesian and widely used:

sofa = sofa / couch

Other options:

kursi sofa = sofa chair (emphasizing it’s a kind of chair)
kursi panjang = long bench (not always a soft sofa)

In ordinary conversation, sofa is the most natural word, so di balik sofa is perfectly normal.

Can I drop saya and just say Kucing bersembunyi di balik sofa?

You can, but the meaning changes slightly.

Kucing saya bersembunyi… = My cat is hiding…
Kucing bersembunyi… = The cat / a cat / cats are hiding… (no owner indicated)

If the context already makes it clear whose cat it is, Indonesian speakers might omit saya or use another modifier, for example:

Kucingku bersembunyi… (informal my cat)
Kucing itu bersembunyi… = That cat is hiding…

Is this sentence formal or informal? What about saya vs aku?

The sentence is neutral and can be used in most situations.

saya is a polite / neutral first-person pronoun, used in formal and many informal contexts.
aku is more informal/intimate, used with friends, family, or in casual speech.

So you could also say:

Kucing aku bersembunyi di balik sofa. (very colloquial)
Or more typically: Kucingku bersembunyi di balik sofa.

Your original Kucing saya bersembunyi di balik sofa is safe almost everywhere.

How do I say My cats are hiding behind the sofa (plural)?

Indonesian often does not mark plurals explicitly if context already shows it.
Your sentence could already mean my cat(s).

If you want to make it clearly plural, you have a few options:

Reduplication of the noun:
- Kucing-kucing saya bersembunyi di balik sofa.
  = My cats are hiding behind the sofa.
Use para (more formal, often for people, but can be used stylistically):
- Para kucing saya bersembunyi di balik sofa.

Most commonly, people would just rely on context, or say banyak kucing saya (many of my cats) if they want to emphasize plurality.