As you pointed out, $q_i+Q_{-i}=Q$. Summing over all $i$ yields
$$Q+\sum_i Q_{-i}=nQ.$$
Then sum over all $j$ for the best-response functions:
\begin{eqnarray}
Q&=&\sum_{j}q_j=\frac{n(a-c)}{2b}-\frac 1 2 \sum_{j}Q_{-j}=\frac{n(a-c)}{2b}-\frac 1 2(n-1)Q\\
\Rightarrow Q&=&\frac{n(a-c)}{(n+1)b}.
\end{eqnarray}
From here, you can solve for $Q$, and then get each $q_i$ because
\begin{eqnarray}
q_i&=&\frac{a-bQ-c}{2b}+\frac{q_i}2\\
\Rightarrow q_i&=&\frac{a-bQ-c}{b}=\frac{a-c}b-Q\\
&=&\frac{a-c}b-\frac{n(a-c)}{(n+1)b}=\frac{a-c}{b(n+1)}.
\end{eqnarray}
The discounting multiplies the annual revenue by a factor of $10$. Thus the profit for firm $i$ is $10Q_i(800-Q_1-Q_2)-500Q_i=10Q_i(750-Q_1-Q_2)$. We can drop the factor ten and maximize $Q_i(750-Q_1-Q_2)$.
Given an annual capacity $Q_1$, firm $2$ wants to maximize $Q_2(750-Q_1-Q_2)$. Differentiating with respect to $Q_2$ yields $750-Q_1-2Q_2$, which is zero at $Q_2=375-Q_1/2$.
In Cournot competition, this must in turn lead to firm $1$ choosing $Q_1$, so we must have $Q_1=375-Q_2/2=375-(375-Q_1/2)/2$, and thus $Q_1=250$, and likewise $Q_2=250$.
In Stackelberg competition, firm $1$ anticipates firm $2$'s response $Q_2=375-Q_1/2$ and thus maximizes $Q_1(750-Q_1-(375-Q_1/2))=Q_1(375-Q_1/2)$. The derivative with respect to $Q_1$ is $375-Q_1$, so firm $1$ chooses $Q_1=375$, which leaves $Q_2=375-375/2=375/2$ for firm $2$.
For the monopoly capacity, (one tenth of) the profit is simply $Q(750-Q)$, the derivative with respect to $Q$ is $750-2Q$, so the optimal capacity for a monopoly is $Q=375$, the same as the capacity for the leader firm in Stackelberg equilibrium.
From the consumer perspective, the Stackelberg competition is the best, with a total annual capacity of $562.5$ being sold at a price of $237.5$, followed by the Cournot competition with total annual capacity $500$ and price $300$, and unsurprisingly the monopoly is worst with total annual capacity $375$ and price $425$.
The profits of the firms (without the factor $10$) are $375^2=140625$ for the monopolist, $250^2=62500$ per firm in Cournot competition, $375^2/2=70312.5$ for the leader firm in Stackelberg competition and $375^2/4=35156.25$ for the follower firm in Stackelberg competition. Assuming a $50/50$ chance of being the monopolist or leader firm, the expected profit per firm is $375^2/2=70312.5$ for the monopoly, $250^2=62500$ for the Cournot duopoly and $(375^2/2+375^2/4)/2=3/8\cdot375^2=52734.375$ for the Stackelberg duopoly, so the firms' preferences are in the opposite order to those of the consumers.
Best Answer
A way to understand the difference between Nash equilibrium and subgame perfect equilibrium is the following. In a Nash equilibrium every party is playing a best reply (to others' actions) on the equilibrium path. In a subgame perfect equilibrium, every party is also (planning to) play a best reply off the equilibrium path.
In your example, consider Player 1. The equilibrium path has 1 playing 64 and 2 playing 418. As shown in the slides you mention, each party is playing a best reply. So it is a Nash equilibrium.
We get off the equilibrium path when 1 deviates. Suppose 1 plays $q_1 \ne 64$; f.i., he plays $q_1 = 50$. Then 2's best reply should be $q^*_2(50) = 425$ whereas her plan is to play $q_2 = 1000$. So she is not best replying to 1's possible deviations.