Learn Hardcore Malay: Kadang-kadang jerebu datang dari bandar lain dan menutup matahari. - Sometimes haze comes from another city and covers the sun.

Kadang-kadang literally comes from kadang (occasion, time) repeated twice. Reduplication like this often gives a "from time to time" or "now and then" meaning, so kadang-kadang = sometimes.
The hyphen just shows that it is a reduplicated word. You will see this pattern a lot in Malay: pelan-pelan, baik-baik, budak-budak, etc.

Can kadang-kadang go in other positions, for example Jerebu kadang-kadang datang…?

Yes. These are all grammatical, with only slight differences in emphasis:

Kadang-kadang jerebu datang dari bandar lain dan menutup matahari.
– Neutral; common way to start a sentence with a time adverb.
Jerebu kadang-kadang datang dari bandar lain dan menutup matahari.
– Slightly more focus on jerebu; "The haze sometimes comes…"
Jerebu datang kadang-kadang dari bandar lain dan menutup matahari.
– Less common, feels a bit clunky, but still understandable.

Putting kadang-kadang at the very beginning is the most natural in this sentence.

Is jerebu the same as "smoke" or "fog"? When should I use jerebu?

Jerebu means haze / smog, usually caused by smoke and pollution (for example, from forest fires).
It is not exactly the same as:

asap = smoke
kabus / kabut = fog / mist (more like natural fog)

So if you are talking about the kind of thick, dirty air that makes the sky look grey and hides the sun, jerebu is the right word.

Why is there no word like "the" or "a" before jerebu and matahari? How do we know if it means "the haze" or "a haze"?

Malay does not have articles like a, an, or the. Nouns appear without them, and definiteness is understood from context.
So jerebu can be haze, a haze, or the haze, depending on what makes sense in English.
Here, English would naturally choose the haze and the sun, but Malay does not need to show that explicitly.

Why do we use datang here and not another verb like pergi or berasal?

Datang means to come, so it fits well with the idea of haze coming from another city.

Pergi = to go (movement away from us), so jerebu pergi dari bandar lain would be wrong here.
Berasal dari = to originate from; it sounds more static or factual (e.g. Dia berasal dari Melaka – "He is from Melaka"), not like actual movement.

In this sentence, we imagine the haze coming towards us, so datang dari is natural.

What is the function of dari in this sentence, and how is it different from daripada?

Here dari means from and marks the physical origin of the haze: datang dari bandar lain = comes from other cities.
A common guideline:

dari – for physical place and time: datang dari Kuala Lumpur, dari pagi sampai malam
daripada – for comparisons, sources in a more abstract sense, or after some verbs: lebih baik daripada, belajar daripada pengalaman

In everyday speech people sometimes mix them, but in this sentence dari is the standard choice.

Does bandar lain mean "another city" or "other cities"? How do you show plural in Malay?

Bandar lain can mean either another city (singular) or other cities (plural). Malay usually does not mark plural on the noun.
Plurality is understood from context or added words:

banyak bandar lain – many other cities
beberapa bandar lain – several other cities
bandar-bandar lain – other cities (plural shown by reduplication)

So by itself, bandar lain is neutral: "other city/cities."

Could we say bandar-bandar lain instead of bandar lain? Does it change the meaning?

Yes, you can say bandar-bandar lain, and it is correct.
Bandar-bandar lain clearly emphasizes that there are multiple cities ("other cities").
Bandar lain can be singular or plural depending on context, so bandar-bandar lain is used when you want to make the plurality explicit or sound a bit more formal/emphatic.

Why is it bandar lain, not bandar yang lain?

Both are possible, but there is a nuance:

bandar lain – other city/cities in general, not specifically identified.
bandar yang lain – "the other city/cities", usually referring to a specific set already known in the context.

For example, if you have already mentioned some cities and now you want to refer to the rest of them, bandar yang lain fits better. In a general statement like this sentence, bandar lain is more natural.

What does menutup literally mean, and how is it formed from tutup?

The base word tutup means to close / to shut.
Adding the prefix meN- (here becoming men-) makes it a verb meaning "to perform that action": menutup = to close, to cover (something).
So menutup matahari literally means to close/cover the sun, i.e. to block it from view.

Is menutup matahari a natural phrase? Could I also say menghalang matahari?

Yes, menutup matahari is natural and understandable: the haze covers the sun.
You could also say:

menghalang matahari – blocks/obstructs the sun
mengaburi matahari – makes the sun hazy/blurry (less common)

Menutup matahari focuses on the idea of covering; menghalang focuses more on blocking. All could work, but menutup is simple and common.

How can we tell what dan is connecting in this sentence?

Here dan connects two verbs that share the same subject:

Subject: jerebu
Verb 1: datang dari bandar lain
Verb 2: (jerebu) menutup matahari

So the full structure is: Kadang-kadang [jerebu] [datang dari bandar lain] dan [menutup matahari].
The subject is only stated once, but it applies to both verbs.

What tense is this sentence in? How would I say "Sometimes the haze came..." or "will come"?

Malay verbs do not change form for tense. Datang and menutup are tenseless; time is understood from context or from time words (like kadang-kadang, semalam, esok).
To make time clearer you can add:

Past: Kadang-kadang jerebu datang dari bandar lain dan menutup matahari semalam. (…yesterday)
Future: Kadang-kadang jerebu akan datang dari bandar lain dan menutup matahari.

Adding akan often marks future, but it is optional if the context is clear.

Is matahari always written as one word? Are there other ways to say "sun" in Malay?

Yes, matahari is normally written as one word and means the sun (literally "eye of the day").
There are a few synonyms:

mentari – poetic/literary
Suria – used in formal names or technical contexts (e.g. tenaga suria = solar energy)

In everyday speech, matahari is the standard word.