"She is gone" versus "she has gone"

You should normally use be gone if no direction is specified, have gone with directions:

Where is Cleopatra? She is gone. (= she is away, or dead)

Where is Cleopatra? She has gone to the temple.

This is idiom: it is irregular and only applies to very few verbs. And is gone can still be used with specific directions sometimes, though it is probably rare. The opposite has gone without direction doesn't sound wrong, but it is probably less frequent.

I believe to be + past participle was used to form the present perfect for all intransitive verbs in older English, just as in other Germanic languages: not I have been but I am been, etc. So it was once he is gone always, even with specific directions, as in modern Dutch hij is naar huis gegaan. Then at a certain point in time the default auxiliary verb for the present perfect of intransitive verbs changed to have in English; but some old, very frequent expressions stayed in use, like to be gone and some others.

The word gone here is a predicate adjective, meaning "no longer here". Etymologically, it came from the past participle of go, but in modern English it is also an adjective. Like the adjective ready, the adjective gone cannot usually be placed before a noun. Compare:

The car was packed and ready.
*The ready car was parked at the curb, waiting for them.

When they looked in the cave, the treasure was gone.
*They looked everywhere for the gone treasure.

The Merriam-Webster dictionary does have examples where gone is used before a noun, but in most situations, placing gone before the noun is ungrammatical.

Using is gone is a sentence makes it passive tense, as in ''he is gone by someone's pressure'', while using has gone is active tense. I suppose this is the difference.

She HAS gone to the market

She is gone. (to mean, she left, she's away).

You can also use idioms like "what is gone and what is not...".... that is often used in literature.

Another case in which the verb to be may be used to express the present or perfect tense is demonstrated in this question (now closed) Emily Dickinson poem

A verse is bothering me:"Where is Jesus gone?" (From “Dying! Dying in the night!) Emily Dickinson poem

Christians are brought up reading the Bible: often the King James Version. The King James Version was published in 1611 and was written in a form of English that was even then old-fashioned. As a result, they tend to use archaic language when formally expressing religious thoughts. Emily Dickinson (1830 –1886) was no exception.

The language of KJV1611 shows evidence, and many examples, of the former use of the verb "to be" to form the present perfect and past perfect of intransitive verbs of motion and change.

Because of the Bible’s influence on the language, some of these forms are still used, particularly in dialect and colloquially.

Mic:1:9: For her wound is incurable; for it is come unto Judah; he is come unto the gate of my people, even to Jerusalem.

Ac:21:11: And when he was come unto us, he took Paul's girdle, and bound his own hands and feet,

De:17:14: When thou art come unto the land which the LORD thy God giveth thee,

1Pe:3:22: Who is gone into heaven, and is on the right hand of God; angels and authorities and powers being made subject unto him.

M't:26:71: And when he was gone out into the porch, another maid saw him,

Ge:19:23: The sun was risen upon the earth when Lot entered into Zoar.

Psalms:22:14: I am poured out like water, and all my bones are out of joint: my heart is like wax; it is melted in the midst of my bowels.

Ac:7:40: Saying unto Aaron, Make us gods to go before us: for as for this Moses, which brought us out of the land of Egypt, we wot not what is become of him.

2Ki:4:18: And when the child was grown, it fell on a day, that he went out to his father to the reapers.

Nu:35:32: And ye shall take no satisfaction for him that is fled to the city of his refuge, that he should come again to dwell in the land, until the death of the priest.

For a more modern use:

2009 My Life Journal (from My Heart to Yours) By V. K. Sansone When he is gone to work, I feel lost, like there is a part of me that is missing and since I am "the rib", I guess it is the rest of my body that is gone.

And the Google Ngram for "is gone to,is become"