May $e^+$ and $e^-$ annihilate with two spins so oriented that they add up instead cancel out? May $e^+$ and $e^-$ annihilate with two spins so oriented that they add up instead of cancel out? If this is not possible what science says about the mechanism that stops this from happening? Does something similar happen with two electrons just colliding each other? should they also feel some force acting on their spins while getting close to one another?
 A: There's a heuristic in quantum mechanics that anything not forbidden is compulsory. The process you describe isn't forbidden by any conservation law, so it should happen. Conservation of angular momentum says that the photons emitted in this process will have to have a total spin of 1, aligned in a certain direction.
I believe that in examples like when a $^{22}$Na source emits a positron into matter, the annihilation rate is low enough that the positron usually has time to slow down first and settle into a ground state, which has some half-life. During this process, it seems unlikely to me that the spin would retain any special direction or correlation.
In an accelerator experiment such as an $e^{+}e^{-}$ collision, I'd imagine that if you had polarized beams, the cross-sections would depend on spin.
A: This is a good question, although I am mainly addressing your first two questions. The spin orientation has a large effect on what happens. The electron and positron annihilate to photons. I find it easier to talk about in the context of Positronium, a Hydrogen like "atom" with an electron bound to a positron instead of a nucleus.
Positronium has 2 ground states that differ only in the spin orientation. Both states eventually decay, but to different numbers of photons! The spin-0 state is limited by Charge and Parity selection rules to only decay to final states with even number of photons, whereas spin-1 Positronium can only decay to an odd number of photons.
Since Positronium is an unstable state it has a typical lifetime. If you calculate the lifetimes for spin 0 and spin 1 states you get a very large difference. For one thing each extra photon comes with an extra factor of inverse of the fine structure constant squared. In the end the spin 0 state has a lifetime of 124 ps, and the spin 1 states have 142 ns, over 1000 times longer lifetime. Again with the only difference being the spin orientation.
Similar to Hydrogen there is also "hyperfine structure" that arises from the interaction of the spins. This results in a shift in the energy levels (very small, on the level of meV). This can result in a "spin flip" where the spin 1 state transitions to spin 0. However this is very suppressed and the typical timescale is much longer than the lifetime of the positronium. This is the closest thing to a "force" between the spins. Generally in QFT when a particle exchanges a photon that photon carries both the electric and the magnetic interaction (or at least what we would label as those). You can separate the two effects when you analyze specific scenarios. So in your final question, two electrons exchanging a photon will "feel" both the electric repulsion, but also the magnetic interaction between their mutual spins. The effects scale differently with the energy of the system, with the "magnetic part" becoming more important at higher energies.
A: Electron and positron can annihilate in any spin state; the emitted photons carry away the angular momentum.
One electron can only occupy the same spatial state as another if their mutual spin state is antisymmetric, which is to say (roughly speaking) with oppositely aligned spins. But in a collision they need not ever be in the same spatial state. For example, their momenta can be different. To calculate the overall cross section you have to include all possible avenues for the evolution of the spins, so in this sense the spin state is important.
