The colors most certainly exist without the name. You may have described the fruit as being a weird shade of red, but if someone held up something red and said "so it was this color" you'd say no. Conversely if someone held up something that was actually orange colored, you'd say "yeah it was that color."

Similarly, you may have no idea what the name is for the color of a Tangerine, but you know what that color is. You might describe it as a dark orange. If I say the name for it is coquelicot, you can look up coquelicot and see if it matches the color you picture in your mind.

I don't think so. Just becoming fluent in multiple languages can result in the perception of more distinct colors. And those fluent in languages that have additional distinct color names can differentiate subtle differences in the shades of colors that non-speakers cannot even differentiate. Color is less about seeing what is actually out there and more about how our brain interprets colors to create "meaning".