This is super belated, but I'll try to give an answer to this question, should it be helpful for future homework doers. As is, the question is a bit under-specified because it may not be evident what frequency is being mentioned (resonance, precession, etc., though essentially the same number), which is probably why answers have not been forthcoming.
The short answer is YES, your approach is correct, but I'll make a reasonable assumption of precessional motion being detected, and explain why, based on the assumptions. In this experiment, one uses the Helmholz coil to do the cancellation of Earth's field (plus any other sources), but the magnet is used to provide detection. Basically, a magnet at some small angle with a total net magnetic field $B_T$ will undergo what is known as 'precession' (wobble) about the total magnetic field at a frequency $\omega_p$ proportional to
$\omega_p = 2\pi f \propto \gamma * B_T$
$\gamma$ is called the gyromagnetic ratio and is proportional to charge over mass. $B_T$ is the net field including the Earth's magnetic field, any stray field sources from magnets, and the field produced by the coil current. Your equation above is a form of $(1/2\pi)\gamma * B_T$, and/or the corresponding classical energy, proportional to $\omega^2$. So, when you cancel the earths magnetic field (plus any other stray magnetic field source), it will be observed when the precessional frequency vanishes or becomes zero, or the magnet's energy vanishes (ceases motion). Under these assumptions, your approach is correct :) Hopefully this explains a bit more as to why. So, well done, even though you are probably a Professor by now!
The current direction along the coil depends on the sense of the winding (or thread), so it can be either way. (I can't really make sense of the perspective in your drawings.)
The right-hand rule for coils and their magnetic field implicitly assumes that the current flows on circles around the solenoid axis. In reality, these are of course very slightly bent and form a helix, and the helix can be left-threaded or right-threaded. This does not apreciably change the way the current flows in the angular direction (tangential to the solenoid), and thus the magnetic field. However, the net current direction is reversed.
So if the solenoid is, say, vertical, and magnetic north is up, then the current (direction chosen as for the right-hand rule) is flowing up for a right-winding coil, and down for a left-winding one (if I didn't confuse myself here).
Best Answer
Let's go back and consider the Biot-Savart Law for a filamentary current:
$$B(r) \propto \int_C I(r') \times \frac{r-r'}{4\pi |r-r'|^3} \, dr'$$
As I've written it, $r'$ is some position on the coil (we integrate over the coil to get the whole magnetic field). $r$ is the position we want to find the field at--somewhere inside the coil's empty body.
Let's consider $r$ directly at the center of the coil. As we start integrating, we slide along the coil. I'll traverse the loop clockwise, so let's have $r'$ start at the top of the batter, come across the top of the page, and start traversing the topmost loop of the coil.
As we start on the rightmost part of the topmost loop, the instantaneous direction of current is out of the page. If we take $r = 0$, so that the center of the loop is the origin, then all we have is the vector $I(r') \times [-r'/|r'|^3]$ as the integrand. $-r'$ points inward toward the center of the coil. It should be clear that the resulting magnetic field from a small piece of the wire at this point in the coil is both downward and to the left.
Let's consider what happens when we get to the leftmost part of the topmost loop. The vector $r'$ is downward and to the left. The current is into the page. The resulting magnetic field is downward and to the right.
In general, as we traverse the topmost loop, each small piece of wire adds a magnetic field that is (a) downward and (b) pointing out of the coil. (I must remind that we're talking about the magnetic field only at the center of the whole coil right now).
If the topmost loop were rotationally symmetric, we could argue that any components that point away from the central axis of the coil must cancel. The real coil does not have this symmetry, but it's "pretty close" to being rotationally symmetric, and any such real components ought to be small.
All the other loops work basically the same way, contributing only net downward magnetic field when integrated over a whole circular loop.
For some reason, you referred to the magnetic field contribution from different points as being clockwise or anticlockwise. I do not understand this. This coil does not create any kind of closed magnetic loops that can be seen on this scale. The field is more clearly described using fixed directions (into or out of the page, left or right, down or up).
I think it's this reason that you thought "clockwise and anticlockwise should cancel". But you described it more correctly in saying that at both points, there is a downward component; the only thing that can cancel a downward component is an upward component! You were right to say that at points where the current moves left, the field's direction is down and out of the page, and that when the the current moves right the field direction is down and into the page. It's just that only the into/out of page components cancel, and net downward is left behind.