Το κιβώτιο ταχυτήτων είναι χαλασμένο.

Because κιβώτιο ταχυτήτων literally means box of gears/speeds. In Greek, one noun often follows another in the genitive to mean of.

So:

• κιβώτιο = box
• ταχυτήτων = of speeds / of gears

In car-related Greek, ταχύτητες can mean gears, so κιβώτιο ταχυτήτων is the standard expression for gearbox or transmission.

κιβώτιο is a neuter singular noun. In Greek, articles and adjectives must agree with the noun in gender, number, and usually case.

That is why you get:

• το = neuter singular the
• χαλασμένο = neuter singular broken

Compare:

• ο κινητήρας είναι χαλασμένος — masculine
• η μηχανή είναι χαλασμένη — feminine
• το κιβώτιο είναι χαλασμένο — neuter

After είναι, Greek uses a normal adjective that agrees with the subject. There is no special separate form just because it comes after is.

So in:

• Το κιβώτιο ταχυτήτων είναι χαλασμένο

the subject is το κιβώτιο ταχυτήτων, which is neuter singular, so the adjective is also neuter singular: χαλασμένο.

Here χαλασμένο means something like broken, faulty, damaged, or not working properly.

For machinery, it often means the item is no longer functioning correctly, not necessarily that it is physically smashed.

A useful contrast:

• χαλασμένο = faulty, out of order, malfunctioning
• σπασμένο = broken in the sense of snapped, cracked, shattered

So for a gearbox, χαλασμένο is very natural.

A rough pronunciation is:

to ki-VO-tio ta-hi-TI-ton I-ne ha-la-sME-no

A few helpful notes:

• χ is not English h. It is a rough sound, like the ch in German Bach or Scottish loch.
• The stressed syllables are marked by the written accents:
  • κιβώτιο
  • ταχυτήτων
  • είναι
  • χαλασμένο

In normal Greek, you usually need it here. Το κιβώτιο ταχυτήτων είναι χαλασμένο is the natural sentence.

Greek uses the definite article more often than English. If you say Κιβώτιο ταχυτήτων είναι χαλασμένο, it sounds unnatural in ordinary speech unless the style is very telegraphic, like a note, label, or headline.

No, Greek word order is fairly flexible. The original sentence is the most neutral and straightforward:

Το κιβώτιο ταχυτήτων είναι χαλασμένο.

But you could also say:

• Είναι χαλασμένο το κιβώτιο ταχυτήτων.
• Χαλασμένο είναι το κιβώτιο ταχυτήτων.

These alternatives change the emphasis, but not the basic meaning.

είναι is the present tense, third person singular of είμαι, meaning to be.

So here it simply means is.

Yes. In everyday speech, many people also say σασμάν for gearbox / transmission.

So you may hear:

• Το σασμάν είναι χαλασμένο.

But κιβώτιο ταχυτήτων is the standard full term, and it is perfectly natural.

You would say:

Τα κιβώτια ταχυτήτων είναι χαλασμένα.

Notice the changes:

• το → τα
• κιβώτιο → κιβώτια
• χαλασμένο → χαλασμένα

But ταχυτήτων stays the same, because it is already genitive plural in the fixed expression κιβώτιο ταχυτήτων.