They are indistinguishable if they have the same spin or a measurement is done such that their spin is a superposition (like an Sx measurement). In theory though they are distinguishable between each other (i.e. a spin, Sz=+1/2 is distinguishable from an Sz=-1/2) provided, as I said, something like an Sx measurement is not done.
Depends on the context, doesn't it? The total spin of a system is conserved (non-relativistically), so if you have 5 up and 3 down, you'll always have 5 up and 3 down.
But for any interacting electrons they can swap spins, so you can never tell which one is or was which.
So they're 'distinguishable' from the thermodynamic sense since the total number of each spin is conserved,
and you have Fermi-Dirac statistics and all that. But they're not distinguishable in the sense that you can tell one from the other.
Well I feel like you're muddling picture here. In statistical mechanics we have an indistinguishability amongst those of the same spin. Thus, it's essentially, how can you distribute 5 ups amongst 8 spins. However, if we treat this instead with pure quantum with the schrodinger (or Dirac) solution of 8 distinct electrons it's different. Our solutions must be the same under permutation of ups and downs but it's a different situation where we have initial values conditions and such. Of course I could just be talking out of my arse here.
The trick with distinguishable or not is simple: two quantum systems are identical, if their intrinsical attributes are the same. For elementary particles a\ la Wigner, two elementary particles are identical/indistinguishable, iff they have the same mass and eigenvalue of the squared Pauli-Lubanski pseudovector.
Do the protons in a helium nuclei have spin 0 in the ground state? Or does Pauli's exclusion principle only apply to elementary particles, thus allowing the protons to have spin 1?
Is the correct interpretation to say that the protons are made from 3 quarks, so swapping the two protons swaps 3 quarks (u of one with the u of the other, d of one with d of the other) which produces a negative sign since 3 is an odd number of swaps?