I wouldn't attach the handle to the bucket. I'd just place it so it looked right.
But otherwise, you'd have to Edit Nurbs > Attach surfaces.
There isn't a noise modifier that I know of. I generally use a lattice or maybe a Sine modifier.
Bend and stuff can be found in the Animation menues under Deform > Create Nonlinear >
The bucket in the picture seems to be a little narrower at the base than yours. I think the bottle in the picture might also be a little thinner than the one you've made.
Looking like it should be a good likeness, though.