International Trade: Two country trade problem
Part (a) is correct. Your error in (b) is to implicitly assume that relative prices are equal to $1/2$.

The best way to understand the problem is using the Edgeworth box (excuse the sloppy picture; will add a custom one when had the time):



The axes represent the two goods. Their lengths are 1800 (global endowment of food, x axis) and 900 (global endowment of clothing, y axis). Point 3 represents the endowment point, which is (1200,300) for Home and (600,600) for Foreign. As the figure shows, that point is not an optimal allocation. Such optimal allocation is where the MRS for the two countries is the same (point 1), MRS which is equal to the relative price of goods (same for both countries, i.e. they are international prices).

Therefore, this allocation has the following properties:

$$ \frac{H_f^2}{2H_c} = \frac{F_f^2}{2F_c} = \frac{P_c}{P_f}$$

(I replaced $D$ with home and foreign letters, for clarity).

Notice that the ratios of consumption are the same across countries. Yet, before trade, production is such that the ratio of food versus clothing is larger in home than in foreign. Therefore, home will export food and import clothing, and vice versa for foreign. Yet, since imports of one country are the exports of the other country, you get:

$$ \frac{1200 - A}{300 + B} = \frac{600 + A}{600 - B} $$

where $A$ are exports of food and $B$ exports of clothing.

Additionally, you have that the trade balance must be in equilibrium. Therefore:

$$ P_fA = P_cB $$

Here you have all the equations you need to find the solution in terms of final consumption levels and relative prices. Let me know if this is enough. I can guide you further if you need.