I also think that walter's answer to (a) is wrong.

Given any p, profit=pq-c(q)=q((p-a)-bq) with -b positive. This is increasing in q. Thus the firm's profit maximizing q is infinity giving profit=infinity, for any given p.

So we cannot draw supply curve, neither have an equilibrium.

As Mr.Anonymous pointed, solving FOC mechanically just gives you minimun, instead of maximum. You can see this by cheking the SOC.