Το τραγούδι αρχίζει τώρα.

• Το = the definite article the
• τραγούδι = song
• αρχίζει = starts / begins
• τώρα = now

So grammatically, Το τραγούδι is the subject, αρχίζει is the verb, and τώρα is an adverb telling you when.

Because τραγούδι is a neuter singular noun.

Greek articles change for gender, number, and case. Here the noun is:

• neuter
• singular
• the subject of the sentence

So the correct article is το.

Very roughly:

• ο = masculine singular the
• η = feminine singular the
• το = neuter singular the

Not exactly. In Greek, grammatical gender is a property of the noun itself, and it does not always match natural gender or whether something is an object.

τραγούδι just happens to be a neuter noun. You usually need to learn a Greek noun together with its article:

• το τραγούδι
• το σπίτι
• το βιβλίο

That said, the ending -ι often appears in neuter nouns, so it is a useful clue.

Because the subject is it: the song.

The dictionary form is αρχίζω = I start / I begin.
In the sentence, the subject is third person singular, so Greek uses:

• αρχίζω = I start
• αρχίζεις = you start
• αρχίζει = he/she/it starts

Since το τραγούδι is singular, the verb must be αρχίζει.

It is the present tense, third person singular.

In a sentence like this, Greek present tense can often be translated naturally as:

• starts
• is starting
• begins
• is beginning

The adverb τώρα makes it clear that the action is happening now, so English may use either simple present or present progressive depending on context.

It is in the nominative, because it is the subject of the sentence.

One thing that can confuse learners is that for many neuter nouns, the nominative and accusative look the same. So here:

• το τραγούδι can look the same in nominative and accusative
• you tell the case from its job in the sentence

In Το τραγούδι αρχίζει τώρα, it is clearly the subject, so it is nominative.

The word order can change. Greek is more flexible than English.

This sentence is a very neutral order:

• Το τραγούδι αρχίζει τώρα.

But you could also say:

• Τώρα αρχίζει το τραγούδι.
  This puts more focus on now.
• Αρχίζει τώρα το τραγούδι.
  Also possible, though a bit more marked depending on context.

So yes, the meaning stays similar, but the emphasis can shift.

A rough English-friendly guide is:

to tra-GHOO-dhi ar-KHEE-zee TO-ra

A few sound notes:

• γ in τραγούδι is not a hard English g. It is a softer throat sound.
• δ in τραγούδι sounds like th in this, not like English d.
• χ in αρχίζει is a strong sound like German ch in Bach or Scottish loch.
• The stress falls on:
  • τραγούδι
  • αρχίζει
  • τώρα

The accent mark shows which syllable is stressed.

So:

• τραγούδι
• αρχίζει
• τώρα

Greek spelling uses these accents regularly, so they are important both for pronunciation and correct writing.

No. Greek normally does not need subject pronouns unless you want extra emphasis or contrast.

The verb ending already tells you the person and number, and here the subject noun is stated anyway:

• Το τραγούδι αρχίζει τώρα.

If the context were clear, Greek could even say just:

• Αρχίζει τώρα.

That could mean It starts now or It is beginning now, depending on context.

Yes, that is possible in many contexts.

Both αρχίζει and ξεκινά can mean starts / begins, but:

• αρχίζει is the more direct and standard verb for begin
• ξεκινά often has a sense of sets off / starts off, though it is also widely used for start

For a song, αρχίζει is very natural.

In normal standard Greek, you would usually keep the article:

• Το τραγούδι αρχίζει τώρα.

Leaving it out would sound unnatural in an ordinary sentence. Greek uses articles more often than English in many situations, so learners should get used to learning nouns with their article.