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Mountain... Heavy mountains. Things that change over time.

Sky... Blue sky. What your eyes can't see. What your eyes can see.

Rei compares Mountain and Sky, these are other terms for earth and heaven, terrestrial and the celestial, body and soul. One is visible to the eyes and the other isn't. Rei alludes to physicality's transience with the words "mountains. Things that change over time."
Sun... A unique object.

Water... Something comforting. Commander Ikari.

Mountain, Sky, Sun and Water. Rei has listed the four classical elements of Earth, Air, Fire and Water under different names, the elements that compose the universe. Another aspect of these elements are reflected in Qabalah in the Four Worlds, as well as the in the trigrams of the Book of Changes which share the same names.
Flowers... So many of the same... And so many unneeded.

It has been suggested this line refers to the clones in the Dummy Plug Plant.
Sky... Red, red sky. The colour red. I hate the colour red.

During episode 26 Rei uses the image of the setting sun and the red sky again: "Sunset... vanishing life... My wish... I don't like it."
Red is the colour that Asuka exhibits throughout the series.
Water flowing.

Blood... The smell of blood. A woman who never bleeds.

Rei lists Flowers, Sky, Water and Blood, which partake of Fire, Air, Water and Earth respectively.
Man made from red soil.

Man made from man and woman.

City... A human creation.

These three lines are likely in reference to the first four chapters of Genesis, where these events occur in sequence.
Eva... A human creation.

What is a human? A creation of God?

Is man a human creation?

"Man made from red soil" to "Is man a human creation?" may describe a six-fold process of progressive creation that is akin to the Lesser Countenance, making "the woman who never bleeds" Aima.
The things I possess are a life and soul. I am a vessel for a soul.

Following on from the above interpretation the vessel is allotted onto Kingdom, where the soul intersects the vessel.
Entry plug, the throne for a soul.

The Entry Plugs act as a throne for the soul of the Lilim as the Evangelions are the highest forms of life on the planet besides Adam and Lilith.
Throne of the soul, or the seat of the soul, is a term Descartes coined to describe the pineal gland which he believed to be the place where the soul intersects the body. In more modern New Age thought the pineal gland is usually said to be the manifestation of the anja chakra, also known as the third eye or mind's eye.
See this external link for more information.
Who is this? This is me.

"Who is this who is thinking these thoughts?" I think therefore I am, Rei says to herself, yet Rei is not satisfied with this answer and goes on to ask more directly:
At this point we see Rei's left eye, as we see in the opening credits. The left eye is associated with the moon.
See Rei and the Moon for more details.
Who am I? What am I? What am I? What am I? What am I?

I am myself. This object is me.

Rei takes another shot at it, instead of the "I think therefore I am", she now goes for "I am that I am."
Yet Rei mistakes her form that is the result of the bonds between herself and others for her self, she mistakes the veil for the tabernacle, and so goes on to say:
This is the me that can be seen, yet I feel as though I am not myself.

Rei recognises her previous mistake and that the visible "this object is me" isn't her self, and that there's something beyond it, which is her unseen soul. Rei peers past the veil:
Very strange. I feel as if my body is melting.

I can no longer see myself. My shape is fading.

I feel the presence of someone who is not me.

Is someone there, beyond this?