What are some useful ways to imagine the concept of spin as it relates to subatomic particles? The answers in this question: What is spin as it relates to subatomic particles?
do not address some particular questions regarding the concept of spin:
How  are some useful ways to imagine a particle without dimensions - like an electron - to spin?
How  are some useful ways to imagine a particle with spin 1/2 to make a 360° turn without returning to it's original position (the wave function transforms as: $\Psi \rightarrow -\Psi$).
When spin is not a classical property of elementary particles, is it a purely relativistic property, a purely quantum-mechanical property or a mixture of both?
 A: How should one imagine a particle without dimensions - like an electron - to spin?
You don't. If you want to imagine, then you think classically and it is just a particle spinning... Thinking like that doesn't give you any other insight of what spin really is (an intrinsic angular momentum, behaving like an [orbital] angular momentum).
How should one imagine a particle with spin 1/2 to make a 360° turn without returning to it's original position (the wave function transforms as: Ψ→−Ψ)
Just imagine it ... no big deal. Again, classically this is not possible, but quantum mechanically it is.
When spin is not a classical property of elementary particles, is it a purely relativistic property, a purely quantum-mechanical property or a mixture of both?
The spin of elementary particle is a pure quantum mechanical effect. Edit: See @j.c. comment. Relativity also plays a role.
Any other interpretation/calculation requires things like commutator, symmetry properties and group theory.
The parallel between "real spinning" and "spin" (which is just a name) comes from the fact that the spin operator needed to account for properties of elementary particles behaves (= has the same definition, based on commutators) like orbital angular momentum operator. This again comes from symmetry properties of ... nature.
The goal of quantum physics is to provide a way to calculate properties. If you want to calculate or go deeper in the problem, then you don't need this classical interpretation.
A: About this:
What are some useful ways to imagine a spin 1/2 particle making a 360° turn without returning to its original position, i.e. Ψ→−Ψ ?
The Dirac scissors is an example of such objects:

This picture is from the book by Penrose and Rindler "Spinors and space-time." I suggest reading it.
A: It's possible to do correct quantum mechanics without believing that particles get altered by 360 degree rotations. Use the "density matrix" form instead of "wave function".
http://en.wikipedia.org/wiki/Density_matrix
To convert a quantum wave state $\psi(x)$ or $|a\rangle$ to a density matrix, multiply the ket by the bra as in:
$\psi(x) \to \rho(x,x') = \psi(x)\psi^*(x')$
$|a\rangle \to |a\rangle \langle a|$  
Since the bras and kets take complex phases, i.e.
$e^{+i\alpha}|a\rangle \equiv e^{-i\alpha}\langle a|$
the complex phases cancel. This is more general than the -1 you get by a 360 degree rotation, but a factor of -1 is also a phase and so it's also canceled.
In short while the state vectors or wave functions take a -1 on 360 degree rotation, the (pure) density matrices are left unchanged.
A: There's a paper$^{1}$ by Battey-Pratt and Racey with an intuitive model of spin  1/2.  I'm not sure if it related to reality, but is an interesting read and attempt at an intuitive understanding. 
--
$^{1}$ E.P.Battey-Pratt and T.J.Racey, Geometric model for fundamental particles, International Journal of Theoretical Physics 19 (1980) 437-475. 
A: I find it useful to think about different spaces having different "sizes", such that one complete rotation through space "A" requires only 360 degrees of turning, but one complete rotation through space "B" requires 720 degrees of turning, making space B in some sense "larger" with respect to complete rotations.
Spin 1/2 particles live within the larger space B, where 720 degrees is a full rotation.  As objects within that space, the spin 1/2 particles have essentially four "sides" to them, one side per 180 degrees.  As observers, we live in space A, and every (classical) object around us in space A reveals all of it's sides in a single 360 degree rotation.  But when we try to rotate spin 1/2 particles through their four sides, it takes two full rotations of our space to see one complete rotation of their space.
The trick is that a "full rotation" must be a fundamentally different concept than an "amount of rotation".  Full rotations depend on the space, amounts of rotation are invariant across different spaces.
If this explanation makes sense, it is immediately obvious why we shouldn't think of particles as being "dimensionless".  In fact, certain aspects of the particle are more dimension-full than the space we are used to thinking about.
