Learn Hardcore Indonesian: Keluarga kami saling mendukung ketika kami menghadapi masalah. - Our family supports each other when we face problems.

So the sentence assumes the speaker is talking about their own family, and the listener is not part of that family.
If the listener is part of the family, you would say keluarga kita instead.

What does saling mean, and why do we say saling mendukung instead of just mendukung?

Saling means each other / one another and shows that the action is mutual.

mendukung = to support (one direction)
saling mendukung = to support each other (mutual support)

So:

Keluarga kami mendukung kami. = Our family supports us.
Keluarga kami saling mendukung. = Our family members support each other.

In your sentence, saling explains that the support goes both ways among family members, not just from one person to another.

Where does saling go in the sentence? Can it go after the verb?

Saling comes before the verb:

saling mendukung ✅
mendukung saling ❌

The typical pattern is:

saling + verb

Examples:

Mereka saling mencintai. = They love each other.
Kami saling membantu. = We help each other.

What is the base form of mendukung, and how is it formed?

The base word is dukung (support).
Mendukung is formed by adding the prefix meN-:

meN- + dukung → mendukung

The N in meN- changes according to the first letter of the root; with d, it becomes men-.

Dukung vs mendukung:

dukung can appear in some fixed expressions or more casual speech.
mendukung is the standard active verb form, especially when there is an object:
- Kami mendukung kamu. = We support you.

Why do we repeat kami: keluarga kami … ketika kami menghadapi masalah? Can we drop the second kami?

The sentence:

Keluarga kami saling mendukung ketika kami menghadapi masalah.

literally: Our family supports each other when we face problems.

Repeating kami makes the subject of both clauses very clear.

You can drop the second kami if the subject is obvious:

Keluarga kami saling mendukung ketika menghadapi masalah.

This is still natural. The subject we is understood from context (= the family members).
So both versions are acceptable; the original is just a bit more explicit.

What is the difference between ketika, saat, and waktu here? Could I say saat instead of ketika?

In this sentence:

… ketika kami menghadapi masalah.
= when we face problems.

You can replace ketika with saat with no real change in meaning:

ketika = when (neutral, common in both spoken and written Indonesian)
saat = at the time when / when (also common, slightly more “formal” feel in some contexts)

So you can say:

Keluarga kami saling mendukung ketika kami menghadapi masalah. ✅
Keluarga kami saling mendukung saat kami menghadapi masalah. ✅

Waktu can also mean when, but it’s a bit more colloquial in this function:

Keluarga kami saling mendukung waktu kami menghadapi masalah. (more informal)

Why use ketika instead of kalau? Doesn’t kalau also mean “when”?

Yes, kalau can sometimes mean when, but its core meaning is if (conditional).

ketika / saat focus on time: at the time when
kalau often implies condition: if / when(ever)

In your sentence:

ketika kami menghadapi masalah = at the time when we face problems

If you say:

Keluarga kami saling mendukung kalau kami menghadapi masalah.

it’s understandable and used in casual speech. It sounds a bit like:

“Our family supports each other if / whenever we face problems.”

So ketika is the safest and most neutral choice for straightforward “when” here.

What does menghadapi literally mean, and why is it used with masalah?

The base word is hadap = to face (as in face something in front of you).
With the prefix meng-, it becomes menghadapi = to face / deal with / confront (usually problems, situations, challenges).

So:

menghadapi masalah = to face / deal with problems

Alternative ways to talk about problems:

punya masalah = to have a problem
ada masalah = there is a problem / to have a problem (contextual)
mengalami masalah = to experience problems

Your sentence emphasizes the idea of confronting or dealing with problems together.

Does Indonesian show tense here? How do we know if it means “when we face”, “when we are facing”, or “when we faced” problems?

Indonesian verbs do not change form for tense. Time is usually understood from:

time words (e.g. kemarin, besok)
context
sometimes aspect markers (like sedang, sudah, etc.)

The phrase:

ketika kami menghadapi masalah

can mean:

when we face problems
when we are facing problems
when we faced problems

depending on context. If you want to be more explicit:

ketika kami sedang menghadapi masalah = when we are in the middle of facing problems
ketika dulu kami menghadapi masalah = when we used to face problems / when in the past we faced problems

Is keluarga singular or plural? How does it work with saling?

Keluarga is grammatically singular, meaning family as one unit.
However, it refers to a group of people, so semantically it’s plural.

Keluarga kami = our family (all the family members)

Using saling with keluarga kami is natural because saling refers to the people inside the group supporting each other:

Keluarga kami saling mendukung
= The members of our family support each other.

Can we say kami saling mendukung dalam keluarga kami instead? Is that natural?

You can say:

Kami saling mendukung dalam keluarga kami ketika kami menghadapi masalah.

It’s grammatically correct, but a bit heavier and more repetitive (kami … keluarga kami … kami).

The original:

Keluarga kami saling mendukung ketika kami menghadapi masalah.

is more natural and smoother, because keluarga kami is a nice, compact subject, and everything else describes what this family does.

Can I use satu sama lain instead of saling?

Yes, satu sama lain also means each other.

Keluarga kami saling mendukung.
Keluarga kami mendukung satu sama lain.

Both are grammatical and understandable.
Differences:

saling + verb is more compact and very common.
satu sama lain is a bit more emphatic or explicit and slightly more formal-sounding in many contexts.

In everyday speech, saling mendukung is usually preferred.

Is this sentence formal or informal? Can I use it in both spoken and written Indonesian?

The sentence:

Keluarga kami saling mendukung ketika kami menghadapi masalah.

is neutral in style:

Polite and clear enough for written Indonesian (e.g. an essay, a speech, a school assignment).
Natural and common in spoken Indonesian as well.

So you can safely use it in almost any context.