Answers to first part (stability):

Note that  trace = f'(x) - e  and  det = e ( g'(x) - f'(x) )
(where 'x' is the steady state shown in the picture).

So, taking into account whether f'>0, g'>0, g'(x) > f'(x), etc, we can say:

(a) stable
(b) stable
(c) unstable
    but we know more: if also 0 < e << 1 then f'(x)-e > 0  (because f'(x)>0)
    so we have an unstable node or unstable spiral, that is to say, we have a
    "repelling" point (as required for Poincare'-Bendixon)
(d) saddle