Was Adam really "alone" if God took 'Woman out of Man'? Is a woman alone if out of her womb 'a new one' comes?

The story in Gen. 1 gives a very different account from Gen. 2. This is actually stipulated in the text. After the Gen. 1 story concludes, the Garden of Eden story begins:

This is the account of the heavens and the earth when they were created, when the Lord God made the earth and the heavens. (Gen. 2:4)

(Note: the above can also be read as a summary of Gen. 1, but either way, it serves to distinguish the two accounts.)

The two accounts

In Gen. one there is no "Adam," there are two unnamed human beings, male and female, created in God's image. They are created on the Sixth Day after the vegetation and animals. In Gen. 2, Adam is formed from the dust prior to the vegetation, and Eve is not yet mentioned.

Now no shrub had yet appeared on the earth and no plant had yet sprung up, for the Lord God had not sent rain on the earth and there was no one to work the ground, but streams came up from the earth and watered the whole surface of the ground. Then the Lord God formed a man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being.

I point this out because I think these are two different stories by two different authors and it is a misstep, hermeneutically, to combine them as if they were one. For a summary of the basis for the idea of dual authorship see Documentary Hypothesis. (Note: this does not preclude IMO that both stories were inspired by God, only that it is a mistake to treat them as if they are one account.)

Was Adam alone?

Since the OP asks, however, I will point out that Adam as BOTH male and female is a very old concept. An article in the Jewish publication Forward points out:

The rabbis understood this to mean that Adam was created as an intersexed being, a hermaphrodite; singular in one respect, plural in another. Exactly how Adam was constituted as an intersexed being was debated. Rabbi Jeremiah ben Elazar held that Adam was an androgyne, while Rabbi Samuel bar Nachman held that Adam was more like conjoined male/female twins.

So if Rabbi Samuel is correct, Adam was not alone. However, the text itself does not indicate this. In fact it depicts God as saying "It is not good that then man should be alone."

As long was were are entertaining speculation, there is also a Jewish legend that Adam had a previous wife, before Eve, namely Lilith. She did not like playing a subordinate role, either sexually or otherwise, and left him. If we include this legend, Adam was currently "alone," but he had not always been so. However, once again, this has to be considered speculation.

To take the OP question on directly: Adam was not alone in Gen. 1, but he was alone in Gen. 2, whether Eve was part of him or not.

It would be meaningful to have a look at Gen 1:21-22:

And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that it was good. And God blessed them, saying, Be fruitful, and multiply, and fill the waters in the seas, and let fowl multiply in the earth

And go on to read Gen 1:27-28:

So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.

So, what is meant by being alone is that Adam initially had no female companion through whom he could bring forth his progeny -- until Eve was created. Of course, he could have remained immortal and single. But, having been a creation of bones and flesh, he would have to return to the mud from where he had come (Gen 3:19), so that the progeny would take over through a continuous mechanism of invigoration.

I'm a bit perplexed by this question, but here goes ...

Genesis 18, 21-22

"Yahweh God also said, “It is not good for the man to be alone. I will make for him a suitable helper.” [...] 21 So Yahweh God caused the man to fall into a deep sleep, and while he slept, He took one of the man’s ribs and closed up the area with flesh. 22 And from the rib that the Yahweh God had taken from the man, He made a woman and brought her to him."

The OP asks

"Was The First Man Really 'ALONE' ... if God Took The Woman Out of The Man's side?"

It's pretty clear from the Genesis quote that what is taken out of the man's side isn't a woman, but a rib. That rib then was made into a woman.

Regardless, Adam's rib wasn't a 'suitable helper', it was a rib. It is in this sense that Adam is alone (when Eve is created, there are other creatures as well as Adam, but he is still alone in the relevant sense).