(cont'd): Also does the result on the polynomial interpolant relate to the "exact" Chebyshev coefficients, or the "approximate" and "easier-to-compute" coefficients? (These and other unsure points on the Chebyshev method have made me prefer other methods to approximate functions, especially those that don't introduce transcendental functions.)

I remind you of a revised version of my question on another MathOverflow page. Also: You write that this answer's result "should be easy to map ... to [0, 1]". I don't see how, especially since not every function meeting the result's assumptions on [0, 1] can be extended to one that does so on [-1, 1].