Above is the question and answer to question (b) (ignore (a))... I don't get where the final implication comes from. Why can we use c as a (covariant) index on the LHS... surely we must use d or something different to c. Is the implication obvious, or do we have to justify all the possible combinations in order to understand the implication?
On
You can use the same dummy index on both sides of an equation; the Einstein summation convention applies on each side separately.
The only thing that happens in the last impliciation is that the left-hand side is rewritten using $\mathbf e_a\times \mathbf e_b = \epsilon_{abc}\mathbf e_c$ (which is close to being a definition of the cross product).
Since each side is vector valued, It doesn't matter what you call the index. For instance, say you called one $c$ and the second $d$, you still would have that: $$LHS_{c1}=LHS_{d1},~ LHS_{c2}=LHS_{d2}, ~LHS_{c3}=LHS_{d3}$$ So it these are essentially the same.