How many colours are traditionally counted in a rainbow?

Seven, and the seventh is there for reasons that have nothing to do with optics. A rainbow is a continuous spectrum with no divisions in it at all; where you draw the lines is a decision about language. Newton originally listed five, then added orange and indigo to bring the count to seven, because he wanted the colours of light to match the seven notes of the musical scale and the seven known planets.

Indigo is the one that gives schoolchildren trouble, because most people cannot reliably pick it out as separate from blue and violet. It survives in the mnemonic entirely because of Newton's numerology.

The bow itself is made by light entering a raindrop, bouncing once off the back and coming out again, bent slightly differently for each wavelength - which fixes the angle at about 42 degrees from the point directly opposite the sun. That is why the bow is always a circle centred on the shadow of your own head, why you can never reach the end of it, and why no two people ever see quite the same rainbow. A second, fainter bow sometimes appears from light that bounced twice, with its colours reversed.