How do I interpret the results of this chromatogram and does it tell me about the compounds' polarities?
You have several queries. First of let us look at the gradient method used by the authors.

The HPLC gradient for synthetic colorants was 0 min A 100 p. cent, 10
min A 0 p. cent, 14 min A 100 p. cent. The HPLC gradient for natural
carmine was 0 min A 80 p. cent, 20 min A 50 p. cent, 23 min A 30 p.
cent, 25 min A 30 p. cent

Factually, notice the trend in the mobile phase for synthetic ones. It is a linear gradient elution that they started with 40% MeOH at t=0 min, and at t=14 min, you have 70% MeOH. In the second case, it is a purely empirical &quot;step&quot; gradient. Try to draw a graph on a piece of paper.
Regardless, the authors use a water-rich mobile phase in the beginning and go to methanol rich mobile phase. Note I am avoiding the word polarity.

Polarity is an umbrella term in chromatography. For elementary purposes, it may be okay for serious work predicting retention on the ill-defined polarity that carries little or no meaning. You linked a GC webpage for an HPLC separation; those are different worlds.
Qualitatively, in top figure (a), it is clear that amaranth spends more time in the water-rich mobile phase, followed by azo ruby and then ponceau 4R if we use a C18 phase.

All you can say is that amaranth has a higher solubility in the water-rich mobile phase than ponceau 4R. One should not over-interpret retention behavior with this limited data.
In short, polarity is not a great terminology for describing the retention order. If you are interested, see the &quot;Hydrophobic subtraction model&quot; of Snyder et al. This one describes the retention in a better way of C18 molecules.
2)

In the chromatogram above (specifically chromatogram &quot;a&quot;) and the
associated table, the elution time for the three compounds ranged from
6.60 minutes to 9.76 minutes. Since solvent A and solvent B were used, how do I know which compounds were eluted by which solvent?

This is an interesting question. As you noticed, the composition of the mobile phase changing with time, so the k, the retention factor is changing with time. People have worked out the theory of simple linear gradients. You know the retention time of elution, and you have the gradient composition chart. Correlate it, but this is a very, very rough estimate. The actual treatment is way more involved. Check &quot;Theory of Gradient Elution Liquid Chromatography with Linear Solvent Strength: Part 1. Migration and Elution Parameters of a Solute Band&quot; by Leonid Blumberg to see the complexity of the problem of gradient elution.
One of the best books is  &quot;Introduction to Modern Liquid Chromatography By Lloyd R. Snyder, Joseph J. Kirkland, John W. Dolan, 2011.&quot;