Why can we 'make' new colors simply by adding wavelengths? White light is always said to contain all the different wavelengths of light. Why, then, can we 'make' new colors simply by adding wavelengths? Is it just a matter of our perception, that, when two colors are added together, they appear to be the wavelength of another color. Or are the waves actually forming into another, differently 'waving' wave, with a different wavelength? 
 A: It is simply a matter of perception.
Human eyes have four types of receptor


*

*"cones" that come in three varieties each with a response curve that peaks
at a different frequency (but which have considerable spread and overlap)

*"rods" that work well in lower light than cones do
and effectively provide monochrome vision at night


So, to understand human colour perception in daylight, you only need to work with a model of light as a mixture of different frequencies which pass through one another losslessly.
If a particular source of light is strong at frequencies that mainly stimulate two types of cone, your brain interprets this as an intermediate "colour" between the peaks of each types response curve.
See Wikipedia - Color Vision
Note that TV screens exploit the specific mechanisms of human colour perception. They don't really need to produce an arbitrary range of frequencies, they just need to produce three specific colours at the right mix of intensities to stimulate your receptors to the same degree that an intermediate frequency would.
A: In my answer to this question, I explain how white light is composed of all of the wavelengths output by the Sun in the visible spectrum. I also go on to explain how we can add specific wavelengths to "simulate" white light due to the way our eyes have evolved. The specialized cone cells in our eyes are tuned to different wavelengths and by stimulating them at specific ratios, we can simulate any visible wavelength. I recommend giving it a read as it should perfectly answer your question and it has some really nice and colourful pictures.
A: 
White light is always said to contain all the different wavelengths of light

Well, not really.
The way we perceive colours as a bit complex and explained in details in answers to this question. To summary what colour one is going to see depends on the wavelengths reaching eye retina but also the way our mind interprets them. Some combinations of wavelengths produce white colour in our mind and the one containing all wavelengths (with more or less the same strength on each length) will be interpreted by our mind as white. The truth is there is hardly any source producing all wavelengths. Sun is no exception here, however it has almost full coverage. In the below picture you can see there are some black lines indicating wavelengths missing in the solar spectrum.

Not to repeat entire answer from the referenced question if cones on your retina react the same way as they do for a full spectrum white line your mind will have no way to distinguish it so you'll also see this as a white light. And due to the way our eyes are build you can reproduce the very same effect with a very limited spectrum, in the most narrow version using just 3 wavelengths.
A: Input light has (mathematically speaking - ignore quantum mechanics) an infinite number of possibilities, represented by the spectrum: for each wavelength you tell exact light intensity, and any spectral curve is valid (sharp lines for lasers, fuzzier lines for LED and fluorescent, smooth with occasional dark bands for thermal sources,...).
The eye (or a camera) reduces all this information into 3 intensities: red green and blue. This reduction of information immediately tells you that there are infinite possible spectra that result in the same perception. As a result, white light, or any other light can be produced by combining almost any mix of wavelength, as long as the relative intensites detected by each of the color receptors match what we are trying to reproduce.
On the other hand, RGB space is still 3-dimensional. If you NARROW your available spectra to single wavelengths (sharp spectral lines), you only have a single degree of freedom: the wavelength. By varying the wavelenth, you can move the perceived colour along a curve in 3D RGB space, but it's still just a single curve. You miss most combinations. This is why combinations of pure spectral colours can result in perceived colours that do not match any single wavelength. The best examples are white, magenta and all sorts of desaturated brownish grayish colours.
Because you can simply multiply each colour by a factor to make it brighter, scaling everything in RGB space doesn't matter, so you effectively only have 2D space for describing hue.
This is all very compactly represented in CIE colour space.
