Interpretation of utility function
Yes, $\Pi^n_{i=1}$ refers to the product of a sequence.

Regarding Lucas' solution, the Lagrangian of the problem is:

$$ L (c_1,...,c_n) = U(c_1,...,c_n) - (c-\sum c_i) $$

The FOC with respect to $c_i$ and $c_j$ are:

$$ \frac{\partial L}{\partial c_i} = \frac{\partial U}{\partial c_i} +1 = 0 $$

$$ \frac{\partial L}{\partial c_j} = \frac{\partial U}{\partial c_j} +1 = 0$$

From here you get (the MRS):

$$\frac{\partial U}{\partial c_i} = \frac{\partial U}{\partial c_j} $$

Since

$$ \frac{\partial U}{\partial c_i} = \left(\frac{c_i}{\alpha_i}\right)^{-1} \Pi^n_{i=1} \left(\frac{c_i}{\alpha_i}\right)^{\alpha_i} = \left(\frac{c_i}{\alpha_i}\right)^{-1} U(c_1,...,c_n) $$

from the MRS you get that:

$$ \frac{c_i}{\alpha_i} = \frac{c_j}{\alpha_j} $$

Using these, the definition of $c=\sum c_i$, and the fact that $\sum \alpha_i =1$, you can obtain Lucas's solution (I leave the last bit of the solution to you).

Finally, regarding weights, I think that is just to make the model interesting, as it means consumers might prefer some colors more than others (they spend $\alpha_i$% of their income in colour $i$). If not, they would spend $1/n$ of their income on each, to which any loyal disciple of William of Ockham would rightly observe "they why to bother with multiple colours at all"?