Yes, the mixing is happening in the eyes & brain; no, an RGB mix of yellow isn't the same as a pure yellow frequency; but our eyes will see it as the same.
The eyes have 3 (or 2, if you're colour-blind) types of colour sensors, each of which responds with a different signal profile - each peaks at a particular frequency, and trails off for frequencies that differ from that. The brain merges the signals from those 3 (or 2) different sensors, to make sense of the colour signals to create a single colour signal, and it can't tell whether that was a balanced combination of red and green, or a pure yellow frequency.
See also this answer to a previous, related question.
That explains most colours we see. Except for when we see a combination of red and blue, with no signals in between. There isn't a colour in the spectrum for that - the colours in between red and blue all feature higher signals in the middle, around green. To have signals from red and blue but not green, doesn't map to the spectrum. And our brain won't show a combination of two or more colours for a single point, it always maps a single point to a single colour.
So our brain creates a new colour, not on the spectrum, for a combination of red and blue. Hence, purple pigments aren't real, in that sense - purple is the brain's interpolation of red + blue + no green. Purple is just a pigment of our imagination.
The answer to your question is the obverse of it: we assign a color to an object based on the wavelengths which are reflected to our eyes (or in the case of filters, transmited to our eyes). That means other wavelengths are absorbed. The absorption of wavelengths is based, primarily, on the chemistry of the object.
Red dye applied to cotton cloth is a chemical whose molecules absorb less red light than other wavelengths, hence the red wavelengths are more intense than other wavelengths in comparison to the light from other objects. Similarly for blue, green, yellow, etc objects. Most objects of colors don't absorb all the energy of other wavelengths; they just absorb less of certain wavelengths, and we assign a color name based on the modified mixture reaching our eyes.
In fact, the "colors" surrounding each other can modify our interpretation of what color we see. (Search for "color optical illusions". There are fascinating examples.)
Regarding absorb and reflect: they mean exactly what you think. The energy of an EM wave is taken into a molecular structure and not released as the same wavelength (absorption) or it is released as the same wavelength (reflection or transmission).
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LCD displays work by absorbing light, which is why they need a backlight (or a reflective surface) behind them. I think their default state is to not absorb light i.e. to make the screen black requires power, though I wouldn't swear to this, so when you're looking at an LCD display that is off, you're looking straight through the liquid crystals and seeing whatever colour the backlight is when it's off.
I'm not sure what you're asking re the behaviour in sunlight. When powered off or set to white the liquid crystal doesn't absorb light, so the liquid crystal doesn't heat up but whatever is behind it will heat up. When the LCD is on and set to black it absorbs light, so the liquid crystals will heat up but the screen behind them won't.
LED screens are completely different. They use light emitting diodes to generate light. I don't think light absorption by a light emitting diode greatly affected by whether it's on or not, so they will heat up at roughly the same rate whether they are on or off. Note however that LEDs generate heat, when powered on, due to resistive heating and unless they're in unusually strong light the majority of the temperature rise will be from resistive heating.