Profit maximizing, Quantity and Price with Dominant Firm and Competitive Fringe
Your answer for part (a) is correct. 

For part (b) you need to first determine what's called 'residual demand', which can intuitively be thought of as 'leftover' demand given how much the competitive fringe will produce. Denote residual quantity demanded as $Q_r$. 

Marginal cost for the competitive fringe is:
$$MC_f=\frac{d}{dQ_f} C_f = \frac{Q_f}{2}.$$
To find the supply function for the competitive fringe set $MC_f=P$
$$Q_f=2P.$$

To find residual demand, subtract fringe supply from direct market demand (not inverse market demand). Direct market demand is given by 
$$Q=1005-P$$
so residual demand is
$$Q_r=Q-Q_f=1005-P-2P$$
$$Q_r=1005-3P.$$

Rearranging residual demand gives inverse residual demand of 
$$P=335-\frac{Q_r}{3},$$
which gives residual marginal revenue of
$$MR_r=335-\frac{2Q_r}{3}.$$
First find the profit-maximizing output for the leader firm by setting $MR_r=MC$:
$$335-\frac{2Q_r}{3}=5$$
$$Q_r=495.$$
To find the leader's profit-maximizing price, plug $Q_r$ into inverse residual demand to get
$$P=335-\frac{Q_r}{3}=335-165=170.$$

Finally, we can find the quantity supplied by the fringe by plugging the leader's price into the fringe supply function
$$Q_f=2P=2*170=340.$$