Zij begint te huilen en te niesen, want ze is verkouden en moe.

It can mean either one in Dutch, but the verb tells you which it is.

• zij begint = she begins
• zij beginnen = they begin

Because the sentence has begint (singular), zij here means she.

They mean the same thing here: she.

The difference is mainly stress and style:

• zij is the full/stressed form
• ze is the unstressed/reduced form

At the beginning of a sentence, zij is quite natural, especially in writing. Later in the sentence, ze sounds more neutral and conversational. So in this sentence, both refer to the same person.

Because the verb has to agree with the subject.

The infinitive is beginnen = to begin.
For she in the present tense, Dutch uses begint.

So:

• ik begin = I begin
• jij begint / begin jij = you begin
• hij/zij begint = he/she begins
• wij/jullie/zij beginnen = we/you(pl)/they begin

Since the subject is singular zij = she, begint is correct.

After beginnen in the sense of start to do something, Dutch normally uses te + infinitive.

So:

• beginnen te huilen = to start crying
• beginnen te niesen = to start sneezing

This is a very common pattern in Dutch:

• Hij begint te lachen.
• Wij beginnen te werken.

So begint te huilen literally works like begins to cry.

Because both verbs belong to begint, and Dutch often repeats te before each infinitive in a pair.

So this is very natural and standard:

• te huilen en te niesen

You may sometimes hear or see the second te omitted in looser usage, but for a learner, repeating it is the safest and clearest choice.

Because want is a coordinating conjunction, not a subordinating one.

After want, Dutch keeps normal main-clause word order:

• want ze is verkouden en moe

Compare that with omdat, which does send the verb to the end:

• omdat ze verkouden en moe is

This is a very common learner question:

• want → normal order
• omdat → verb goes to the end

Because they come after is and are being used as predicate adjectives.

In Dutch, adjectives usually stay uninflected after verbs like zijn, worden, and blijven:

• Ze is moe.
• Hij is ziek.
• Ik ben verkouden.

Adjective endings are more relevant before nouns, for example:

• de verkouden man
• het moeë kind (less common-looking, but possible in the right context)

Here, since the words come after is, plain verkouden and moe are correct.

Yes, and this is an important difference between the languages.

Dutch commonly expresses this idea with zijn + verkouden:

• Ik ben verkouden.
• Zij is verkouden.

Literally, that looks like I am cold-ish / she is cold-ish, but the actual meaning is to have a cold.

Dutch can also use een verkoudheid in some contexts, but verkouden zijn is the everyday, natural way to say it.

Normally, no. Dutch usually needs an explicit subject in a new clause.

So after want, you need:

• want ze is verkouden en moe

You would not normally drop ze the way some languages can drop pronouns. Dutch generally behaves more like English here: the new clause needs its own subject.

In standard writing, a comma before want is very common and usually expected when it introduces another main clause.

So this punctuation is normal:

• Zij begint te huilen en te niesen, want ze is verkouden en moe.

The comma helps show that the second part gives the reason or explanation.