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.
There is a standard way of solving for $Q_1$ and $Q_2$.
Determine the profit functions.
Determine the best response function for the firms.
Substitute $Q_1$ or $Q_2$ in the other profit function and solve.
All these steps are already mentioned, so you know what to do. Below you can search for your mistake.
The profit function for firm 1 equals $\Pi_1= P_1Q_1-C_1=Q_1 \cdot (100-0.5(Q_1+Q_2)) - 50Q_1$
The profit function for firm 2 equals $\Pi_2=P_2Q_2-C_2=Q_2 \cdot (100-0.5(Q_1+Q_2)) - 24Q_2$
The best response function can be determined by deriving the profit function of firm 1 w.r.t. $Q_1$ and for firm 2 w.r.t. $Q_2$ and set them equal to zero
$$\frac{\partial \Pi_1}{\partial Q_1}=100-Q_1-0.5Q_2-50=50-Q_1-0.5Q_2=0$$
$$\implies Q_1=50-0.5Q_2$$
$$\frac{\partial \Pi_2}{\partial Q_2}=100-Q_2-0.5Q_1-24=76-Q_2-0.5Q_1=0$$
Now we can make the substitution
$$76-Q_2-0.5 \cdot (50-0.5Q_2)=0$$
$$\implies 51-Q_2+0.25Q_2=0 \implies 0.75Q_2=51$$
And thus we find $Q_2=68$ and can solve easily for $Q_1$
$$Q_2=68 \ \text{and} \ Q_1=50-0.5 \cdot 68=16$$
Best Answer
To answer this question, think about the "vanilla" Cournot competition case, where products $p_1$ and $p_2$ are identical; they're perfect substitutes. In this case, increases in production from your competitor (i.e. $q_2$) displaces your own production, so $d = b$ and
$p_1(q_1,q_2) = a - b(q_1+q_2)$.
On the other hand, if an increase in production of $q_2$ increases demand for your own product $q_1$, then these products are compliments. Be careful about stating they are perfect compliments, because without looking at consumer indifference curves, we can't determine this.
In this case, $d$ is negative, and is bounded by $-b$.
In short, $d$ is a measure of the degree to which these two goods are complements or substitutes. Another approach would be to take the derivative of demand with respect to production of the other good, like this:
$\frac{\partial p_1}{\partial q_2} = -d$.
If $d>0$, $\frac{\partial p_1}{\partial q_2} <0$ and $q_2$ is a complement to $q_1$. Likewise, if $d<0$, $\frac{\partial p_1}{\partial q_2} >0$ and $q_2$ is a substitute for $q_1$. Because of the symmetry of the problem, both will either be complements or substitutes. However, in the real world this is not always the case.
The Cournot-Nash equilibrium is the output {$q_1,q_2$} from which neither firm can profitably deviate. To answer this, you need to find the best response function for each firm by solving for the optimal output, given the production of the other firm. This is accomplished by equating Marginal Revenue = Marginal Cost. Note that the marginal cost of production is zero; i.e. $c'(q_1) = c'(q_2)=0$.
$BR_1(q_2) = \frac{a-dq_2}{2b}$ and $BR_2(q_1) = \frac{a-dq_1}{2b}$.
The Cournot-Nash equilibrium is located where these two Best Response functions intersect. Solving the system of two equations and two unknowns, I get:
$q_1^* = q_2^* = \frac{a(\frac{1}{2}-\frac{d}{4b})}{b-\frac{d^2}{4b}}$.