Learn Hardcore Chinese: wǒ yéye hé nǎinai zhù zài jiāoqū. - My grandpa and grandma live in the suburbs.

With 的: 我的爷爷和奶奶 is a bit more formal/explicit, like “my grandfather and grandmother”.
Without 的: 我爷爷和奶奶 is very common in spoken Chinese for close family members. For kinship terms like 爸爸, 妈妈, 爷爷, 奶奶, 哥哥, 妹妹, etc., 的 is often dropped when the relationship is close and clear.

So the sentence is natural as it is, and you can safely omit 的 with close family words in everyday speech.

Why is it 我爷爷和奶奶 and not 我爷爷和我奶奶?

Chinese tends to avoid repeating the same possessive when it is obvious that both nouns belong to the same person.

我爷爷和我奶奶 is understandable, but it sounds unnecessarily repetitive.
我爷爷和奶奶 is interpreted as “my grandpa and (my) grandma”. The 我 is understood to apply to both 爷爷 and 奶奶.

If the two people did not share the same possessor, then you would need to mark it clearly, for example:

我爷爷和他奶奶 – my grandpa and his grandma

What exactly does 和 do here? Is it like “and” or “with”?

In this sentence, 和 is functioning as “and”, simply linking two nouns:

爷爷和奶奶 – grandpa and grandma
我爷爷和奶奶 – my grandpa and grandma

和 can also mean “with” (indicating accompaniment), as in:

我和爷爷住在郊区。 – I live in the suburbs with my grandpa.

But in your sentence, 和 is just a conjunction joining two people in the subject.

Why do we need both 住 and 在? Could we just say 我爷爷和奶奶住郊区?

The pattern 住在 + place is the most common and natural way to say “live in/at + place” in modern spoken Chinese.

住 means “to live / to reside”.
在 marks the location, similar to “at / in”.

So:

住在郊区 ≈ “live in the suburbs”.

You can say 住郊区 in some contexts, especially in written or more concise styles (like headlines), but for everyday speech, 住在郊区 sounds smoother and more natural.

What is the function of 在 here? Is it a verb “to be at” or a preposition?

In this pattern, 在 is usually analyzed as a preposition (or a coverb) indicating location.

The structure is:

[Subject] + 住 + 在 + [Place]
我爷爷和奶奶 + 住 + 在 + 郊区

You can think of 在 as “at / in” here, introducing the place where the action of 住 happens.

In other sentences 在 can act more like a main verb meaning “to be at / to be in”, for example:

我在学校。 – I am at school.

But with 住, it is part of the 住在 + place construction.

How do we know the tense? Does this mean “live”, “lived”, or “are living”?

Chinese does not mark tense the way English does. 住在 by itself is tenseless. The time is determined by:

Context (what you were talking about)
Time words, like 现在 (now), 以前 (before), 去年 (last year), etc.
Aspect markers, like 了, 过, 着 in other sentences (none here).

So 我爷爷和奶奶住在郊区。 can correspond to:

My grandparents live in the suburbs. (general fact, present)
My grandparents lived in the suburbs. (if the context is in the past)
My grandparents are living in the suburbs (now). (if you are contrasting with some other situation)

By default, with no time words and a verb like 住, many learners interpret it as a general fact, similar to English simple present.

Could we say 我爷爷和奶奶住在郊区里? What does 里 add?

Yes, 我爷爷和奶奶住在郊区里。 is also correct.

郊区 – suburbs / outskirts
郊区里 – literally inside the suburbs

Adding 里 can slightly emphasize “within that area”, but in many cases the meaning is practically the same, and either version is fine. In everyday conversation, most people would just say 住在郊区.

Do we need 们 to show plural, like 我爷爷和奶奶们住在郊区?

No, you should not add 们 here.

爷爷和奶奶 already clearly refers to two people, so plural marking is unnecessary.
们 is mainly used with:
- Pronouns: 我们 (we), 他们 (they)
- Some human nouns to mean a group of that kind of people, e.g. 同学们 (students, classmates as a group).

If you said 爷爷们 or 奶奶们, it would sound like “grandpas” or “grandmas” in general, not your specific grandparents. That is not what this sentence means.

Does 爷爷 and 奶奶 refer to any grandparents, or specifically to paternal ones?

In standard Mandarin:

爷爷 – your paternal grandfather (your father’s father)
奶奶 – your paternal grandmother (your father’s mother)

Your maternal grandparents (mother’s side) are usually:

外公 – maternal grandfather
外婆 – maternal grandmother

So 我爷爷和奶奶住在郊区。 is normally understood as “My (paternal) grandpa and grandma live in the suburbs.”

Context can sometimes broaden or blur this, but the basic default is paternal.

Are the second syllables in 爷爷 and 奶奶 full third/neutral tones? I see yéye and nǎinai—how are they actually pronounced?

In normal speech, the second syllable of both words is pronounced with a neutral tone:

爷爷: yé·ye (second syllable neutral)
奶奶: nǎi·nai (second syllable neutral)

Pinyin often writes both syllables with a tone mark when listing vocabulary (yéye, nǎinai), but in real pronunciation, the second syllable is light and unstressed.

So you should pronounce:

爷爷 with a clear 2nd tone on 爷, then a light ye.
奶奶 with a clear 3rd tone on 奶, then a light nai.

Can we leave out 我 and just say 爷爷和奶奶住在郊区?

Yes, you can.

爷爷和奶奶住在郊区。 – Grandpa and Grandma live in the suburbs.

In many contexts, if it is clear you are talking about your grandparents, people will understand it that way even without 我. However:

Adding 我 (我爷爷和奶奶) makes it explicit that you are referring to your own grandparents, not someone else’s or just “some grandparents” in a story.

In an isolated sentence for learners, including 我 is helpful to make the meaning unambiguous.

Is the word order here fixed? Could I say 我爷爷和奶奶在郊区住 instead?

The most natural and basic pattern for 住 with a place is:

[Subject] + 住在 + [Place]
- 我爷爷和奶奶住在郊区。

You can say 我爷爷和奶奶在郊区住, but:

It sounds a bit marked or emphatic, as if you are stressing “it is in the suburbs that they live”, often contrasting with another place.
It is more common in spoken language when emphasizing location, for example:
- 他们在市中心上班，在郊区住。
  They work in the city center and live in the suburbs.

For a simple, neutral sentence, 住在郊区 is the best choice.

What exactly does 郊区 mean? Is it the same as “countryside”?

郊区 means the suburbs / the outskirts of a city.

It is near a city, often under the city’s administration, but not the central urban area.
English equivalents: suburbs, outskirts, peripheral districts.

It is not exactly the same as:

农村 / 乡村 – countryside, rural villages
乡下 – colloquial “the countryside”

So 住在郊区 suggests living in the suburban areas, not in a distant rural farming area.