What spinor field corresponds to a forwards moving positron?

Dirac spinors are an infuriating subject, because there are about four subtly different ways to define phrases like "the direction a spinor is going" or "the charge conjugate of an spinor". Any two different sources are guaranteed to be completely inconsistent, and all but the best sources will be inconsistent with themselves. Here I'll try to resolve a tiny piece of this confusion. For more, see my answer on charge conjugation of spinors.

Classical field theory

Let's start with classical mechanics. We consider plane wave solutions of classical field equations, which generally have the form $$\alpha(k) e^{-i k \cdot x}$$ where $\alpha(k)$ is a polarization, e.g. a vector for the photon field, and a spinor for the Dirac field. The momentum of a classical field is its Noether charge under translations, so in general $$\text{a plane wave proportional to } e^{-ik \cdot x} \text{ has momentum proportional to } k$$ Now let's turn to the plane wave solutions for the Dirac field, $$\sum_{p, s} u^s(p) e^{-i p \cdot x} + v^s(p) e^{i p \cdot x}.$$ By comparison to what we just found, we conclude $$\text{classical plane wave spinors with polarization } \begin{cases} u^s(p) \\ v^s(p) \end{cases} \text{ have momentum } \begin{cases} p \\ -p. \end{cases}$$ That is, for Dirac spinors, the parameter $p$ doesn't correspond to the momentum of a classical plane wave solution. However, this doesn't tell us about how a wavepacket moves, because plane waves don't move at all. Instead, we need to look at the group velocity $$\mathbf{v}_g = \frac{d \omega}{d \mathbf{k}}$$ of a wavepacket. For the negative frequency solutions, both $\omega$ and $\mathbf{k}$ have flipped sign, so $$\text{wavepackets built around } u^s(p) \text{ and } v^s(p) \text{ both move along } \mathbf{p}.$$ I think this is the best way to define the direction of motion in the classical sense. (Some sources instead say that $v^s(p)$ moves along $- \mathbf{p}$ but backwards in time, but I think this is not helpful.)

Quantum field theory

When we move to quantum field theory, we run into more sign flips. Recall that in quantum field theory, a plane wave solution $\alpha(k) e^{-i k \cdot x}$ is quantized into particles. To construct the Hilbert space, we start with a vacuum state and postulate a creation operator $a_{\alpha, k}^\dagger$ for every mode.

If we do this naively for the Dirac spinor, the raising operator for a negative frequency mode creates a particle with negative energy. This is bad, since the vacuum is supposed to be the state with lowest energy. But Pauli exclusion saves us: we can instead redefine the vacuum to have all negative frequency modes filled, and define the creation operator for such a mode to be what we had previously called the annihilation operator. This is the Dirac sea picture. Then $$\text{particles made by the creation operators for } u_s(p) e^{-ip \cdot x}, v_s(p) e^{ip \cdot x} \text{ have momentum } p.$$ Moreover, both of these particles move along the direction of their momentum $\mathbf{p}$. All other quantum numbers for the $v$ particles are flipped from what you would expect classically, such as spin and charge, but the direction of motion remains the same because the quantum velocity $\hat{\mathbf{v}}_g = d \hat{E} / d \hat{\mathbf{p}}$ stays the same.

Summary

To summarize, I'll quickly assess your arguments.

1. Your first argument is wrong. Momentum doesn't correspond to propagation direction. You know this from second-year physics: a traffic jam is an example of a wave which moves backwards but has positive momentum.
2. Your second calculation is correct, $v^s(p)$ indeed has momentum $-p$.
3. The classical spinor does move the same direction as the quantum particle; it must, if we can take a classical limit. In both cases the classical/quantum spinor moves along $\mathbf{p}$.
4. Indeed, spin up and down is interchanged by the implicit hole theory argument going on, along with everything else.
5. Tong is generally a great source, but he messed up here. I've emailed Tong and he's agreed and fixed it in the latest version of the notes.

Other sources may differ from what's said here because of metric convention, gamma matrix convention, or whether they consider some subset of the objects to be "moving backwards in time".