- Can be done as...
fn[t_] := {2.9 Cos[3.2 Pi t], Sin[4 Pi t] + 5 t};
ans = NMinimize[{EuclideanDistance[fn[t1], fn[t2]], 0 < t1 < 1 && 0 < t2 < 1 && t1 < t2 - 0.1}, {t1, t2}]
ParametricPlot[fn[t], {t, 0, 1}, Epilog ->
{PointSize[0.05], Point[fn[0]], Point[fn[1]], {Point[fn[t1]], Point[fn[t2]]} /. ans[[2]]}]
And 3. can be done as
Plot[Evaluate[Norm[D[fn[t], t]]], {t, 0, 1}]