Η προξενική αρχή είναι ανοιχτή.

Η is the definite article, meaning the. It is the feminine singular nominative form, because the noun αρχή is feminine singular and is the subject of the sentence.

In Greek, when an adjective comes before the noun, the article usually goes before the adjective, so:

η προξενική αρχή = the consular authority / the consulate

Because προξενική αρχή is a noun phrase made of:

• προξενική = an adjective, consular
• αρχή = a noun

So Greek is literally using an adjective + noun structure, just like English can do in consular authority.

It often does, yes. But αρχή is a word with several meanings. Depending on context, it can mean:

• beginning
• principle
• authority / office

In προξενική αρχή, it means authority or office, not beginning. This is a very common thing in Greek: one word can have several related meanings.

Because both adjectives agree with αρχή, which is feminine singular.

Greek adjectives must match the noun they describe in gender, number, and case.

So here:

• προξενική αρχή = feminine singular
• ανοιχτή = also feminine singular

If the noun were neuter, for example, the adjective would change:

• Το προξενείο είναι ανοιχτό. = The consulate is open.

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

Here it means is, because the subject is singular:

Η προξενική αρχή είναι ανοιχτή. = The consular authority is open.

A useful thing to know: είναι is also the form for they are in Modern Greek, so both singular and plural 3rd person use είναι.

For example:

• Η αρχή είναι... = The authority is...
• Οι αρχές είναι... = The authorities are...

Yes. ανοιχτή is an adjective meaning open.

After είναι, Greek often uses an adjective to describe a state or condition:

• Η πόρτα είναι ανοιχτή. = The door is open.
• Το μαγαζί είναι ανοιχτό. = The shop is open.

So ανοιχτή is not a verb. The verb is είναι.

Yes. Both ανοιχτή and ανοικτή mean open.

In everyday Modern Greek, ανοιχτός / ανοιχτή / ανοιχτό is very common.
ανοικτός / ανοικτή / ανοικτό is also correct and may sound a bit more formal or careful in some contexts.

So you may hear either:

• Η προξενική αρχή είναι ανοιχτή.
• Η προξενική αρχή είναι ανοικτή.

Normally, no. In a full standard sentence, you should keep είναι.

So the normal sentence is:

Η προξενική αρχή είναι ανοιχτή.

You might see shortened or headline-style language on signs or notices, where Greek drops words more freely, but for normal speaking and writing, use είναι.

Yes, Greek word order is more flexible than English. The version you have is the most neutral and straightforward:

Η προξενική αρχή είναι ανοιχτή.

You could also say:

Ανοιχτή είναι η προξενική αρχή.

That sounds more emphatic, as if stressing it is open.

However, the phrase προξενική αρχή should stay together, because it functions as one unit.

A rough pronunciation guide is:

Η = ee
προξενική = pro-kse-ni-KEE
αρχή = ar-HEE or ar-KHEE
είναι = EE-ne
ανοιχτή = a-ni-HTEE

A few helpful notes:

• Greek accent marks show the stressed syllable.
• The stress falls on:
  • προξενική → last syllable
  • αρχή → last syllable
  • είναι → first syllable
  • ανοιχτή → last syllable
• The letter χ is not like English ch in church. It is a throaty sound, similar to the ch in German Bach or Scottish loch.

Because προξενική αρχή is one noun phrase, and the article normally appears once at the beginning:

η προξενική αρχή

Greek does not need a second article here. The single article covers the whole phrase.

This is the normal pattern:

• η μεγάλη πόρτα = the big door
• η προξενική αρχή = the consular authority

So the structure is:

article + adjective + noun