Do electrons in an atom always have the same 'direction'?

As you intuit, it is indeed pretty hard to define a sense of "direction" for an atomic electron within quantum mechanics when the electron doesn't have an orbit but it is instead some fuzzy ball of probability, but it is doable and in fact it is one of the central constructs in atomic physics.

What ends up mattering is angular momentum, i.e. how much the electron's motion "turns" around the nucleus. As it happens, it is perfectly possible to define a fuzzy ball of probability for the electron which does not have a definite position and does not have a definite (linear) momentum, but which does have definite angular momentum. A bit weird, but that's what it is. Most atomic electrons are in states like this.

As to your broader question - whether all the electrons are going around in the same direction or not - the answer is simply "it depends".

  • Some atoms, like the noble gases, the alkaline earth metals, and the zinc-cadmium-mercury right-hand edge of the transition metals, have "full shells" which roughly means that for every electron going clockwise about a given axis there is another electron going counterclockwise.

  • Some atoms, like vanadium, cobalt or nickel, have many unpaired electrons, and have a fairly large overall angular momentum of the electronic motion.

In general, the angular momentum of any closed electron shell is zero (i.e. the electrons in the inner shells have one counterclockwise electron for every clockwise one) and it is the outermost, 'valence' shell that determines the angular momentum properties of the atom.

Yes, quantum mechanics allows you to speak of clockwise or anti-clockwise motion, but it comes with the usual caveats of quantum mechanics. The tool that tells clockwise from anti-clockwise motion is the angular momentum. Motion is anti-clockwise around an axis, say the $z$-axis, if the component of angular momentum along that axis is positive. There is an observable for $L_z$ in quantum mechanics, so this carries over. But it comes with the usual caveat that in quantum mechanics $L_z$ need not have a definite value. (Some people would phrase this as that the motion can be "both clockwise and anti-clockwise at the same time" just like certain cat is supposedly "both dead and alive". I think this is trying to force QM to be CM "under the hood", when in reality it's exactly the opposite.)

As for the degree to which atoms have definite angular momentum: in a hydrogen-like atom, the interaction respects rotation symmetry. Therefore, the angular momentum will be conserved and can have a definite value at the same time as the energy does. The conserved angular momentum is, however, the total angular momentum, which is the sum of orbital angular momentum and the spin of the electron. (The sum can have a definite value even though each term does not.)