As Todd noted in a comment, frequencies combine rather than average. Remember that sound is simply moving air (or other medium) being pushed around by, for example, a vibrating string. If you push on something 3 times a second and someone else pushes on it 2 times a second, you wouldn't see it move only 2.5 times per second! You also wouldn't see it move exactly the same amount 5 times, either, since some of your pushes are going to line up at the same time and result in one bigger movement. And when it comes to sound, you're not just pushing but also pulling — so some of your pushes will cancel out.
This graphic (credit to Alex Basson) illustrates how waveforms combine:
As you can see from the final image, both of the original frequencies are represented — the overall frequency is still that of x, but it has a clear sub-frequency that matches y (which is double that of x, in this case). This allows your brain to "pick apart" the sound and hear all of the notes in a chord — different combinations will make different shapes, all representing their components.
Related post that the above was taken from: How do harmonics work?
As for why notes are "chosen" for a chord — well, any two or more notes form a chord. But the most common are structured around a note and its fifth (seven semitones above) because they form a 2:3 frequency ratio. Other than 1:1, 1:2, or 1:3 ratios (two of the same note separate by 0-2 octaves), this is the simplest ratio (the one with the smallest sides) — which means that, while it has bit of intermingling to sound interesting, the frequencies regularly "line up" and form peaks and troughs together, as above. This is called constructive interference, and when it works well and sounds pleasing it's called consonance.
If you want your sound to be more "interesting", you add in notes that line up less often, giving more twists and turns to the waveform. When this becomes too much and just a jumble of frequencies it tends to be unpleasant, and is called dissonance.
For a mathematical representation of consonance/dissonance, see my answer here.