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"Independent of the rate" would mean that the concentration of that specific reactant/product does not interact with or affect the rate of the reaction; this is by definition what a zero order species does, since it is raised to the power of 0 and becomes a 1 (which does not multiply the rate constant by anything that would change it).
Yes, the answers above describe it well. The way I think if it, zero order reactions aren't affected by the reaction and as a result don't change. Therefore, graphing it would be a straight line with a slope of 0.
A zero order reaction also indicates that the rate of that specific reaction is at its maximum (usually due to the introduction of an enzyme) which is why changing the concentration of the reactant will not have an effect on the rate, but only the duration.
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