Well, I did the (tedious) work to translate your equation into Mathematica:
x[s, t] := (a + f[t] Cos[b s] - g[t] Sin[b s]) {Cos [s], Sin[ s], 0} + {0, 0, f[t] Sin[b s] + g[t] Cos[b s]}
Then the normal is given as above by
e1 = D[x[s, t], s] // FullSimplify;
e2 = D[x[s, t], t] // FullSimplify;
nn = Cross[e1, e2]
which gives a real lengthy result, which becomes even more complicated when dividing this by the Length of nn.
Concerning the mean curvature I am not sure whether the formula given above is applicable (I think yes - change x and y to s and t), but you should look for another method to calculate it. But I guess the general formula will be of no use, because it will be even more complicated.
It may help to specify f[t] and g[t], as done in the example below. But does it make sense to speak of a mean curvature of this surface?
ParametricPlot3D[
Evaluate[x[s, t] /. {f[t] -> t, g[t] -> t, a -> 10, b -> .1}],
{s, -5, 5}, {t, -5, 5}]