Can I take off the screen protector on my macbook nano-texture? Is it harmful to the nano coating?

The nano-texture is microscopic etching of the glass, it is not a coating. Removing the screen protector should have no effect on the screen unless it leaves behind any sort of residue.

Can you identify the screen protector? Make/model? Are there instructions about how to remove it?

Is it okay to use the screen without a protector?

Certainly. I think most people don't use a protector. One of the potential problems with laptop screen protectors is their thickness. The distance between the screen & the keyboard is minimal when the lid is closed; the additional thickness of a screen protector can result in pressure being put on the screen when closed, which ultimatly may lead to the screen cracking.

how can I prevent scratches from dirt or the keyboard?

Keep the screen & keyboard clean. Use a clean, damp microfiber cloth to gently wipe the screen. There are screen cleaning cloths made just for this purpose. I use iKlear chamois cloth (damp) & iKlear terry cloth (dry) to clean my screens. I don't use any screen cleaning liquid, just plain warm water.

The MacBook with nano-texture screen is designed to be used without any sort of screen protection added. Using it as such is not damaging in any fashion.

However, removing a screen protector after having added it may leave behind some residual adhesive "stuff" that could affect the nano-texture etching. Still, I would remove the screen protector anyway because, as you say, it has eliminated the anti-reflection properties that nano-texture provides and reduced the screen brightness.

After you have removed the protection, gently clean the screen per Apple's guidance, and continue to use the computer as it was designed.

How to clean your nano-texture Apple display - Apple Support

<< as it might remove the "Nano coating" because it's very thin coating. >>

Nonsense.

The Apple nano-texture glass screen is not a coating AT ALL -- it is etching on the outer face of the glass itself. There is nothing that can be puled off, provided they did indeed sell you the Mac with the etched-face glass screen.

If they sold you a STANDARD Mac screen, THAT anti-reflective coating is very delicate indeed, and the advice to NOT remove any applied screen protectors is good advice. But if that is the case, they charged you for a far more expensive screen, but delivered a cheaper screen. I.e., they cheated you.

Never cover a Mac's screen with anything. That includes a screen "protector". It will cause damage.

Using a camera cover, palm rest, or keyboard cover on a Mac notebook - Apple Support

"... Leaving any material on your display, keyboard, or palm rest might interfere with the display when it's closed and cause damage to your display."

(emphasis added)

I do not see the point in putting a protector on the Nano-Texture display, to me that defeats the purpose. I carry my MBPs around a lot and don't really notice keyboard marks (they don't touch unless the case is pressed in the middle. If it happens, clean with the Apple cloth as linked above (I bought an extra one).