Does the sun have both types of angular momentum? Does the sun have both types of angular momentum, 1. spin angular momentum 2. orbital angular momentum?
 A: Classically, the total angular momentum of any body with reference to an arbitrary reference point can always be considered to be the vector sum of the spin angular momentum with reference to its center of gravity and the orbital angular momentum with reference to this arbitrary reference point. Thus, in general, the sun has both a spin angular momentum due to its rotation and an orbital angular momentum which depends on the point of reference. 
If you take this point of reference to be the center of gravity of our milky way, then the orbital angular momentum is due to the rotational movement of the sun in our galaxy.If you take this point of reference to be the center of gravity of our solar system, then the orbital angular momentum is due to the movement of the center of gravity of the sun around the center of gravity of the solar system.
A: The sun is rotating on it's axis, rotating about its common center of mass with Jupiter (with the other planets too, but those effects are small compared to Jupiter) and rotating about the center of the galaxy.
Spin is a quantum concept, inspired by the idea of a sphere spinning about its center, but isn't actually the angular momentum associated with spinning material.  One key difference between spin and ordinary angular momentum is that a spin-1/2 particle (like an electron) must go through two revolutions to get back to the original state (a little bit like a Mobius Strip).  
It turns out that neither spin nor orbital angular momentum are conserved on their own, we need both to have a fully conserved quantity.
That being said, if the magnetic field of sun is strong enough (I really don't know if it is) it could polarize the plasma that makes up the sun and result in an overall net spin.  The competing processes are the high temperatures (which would tend to erase any polarization), and strong magnetic fields (which would tend to polarize the plasma).    
A quick search gave this result.
http://ieeexplore.ieee.org/document/7773358/?reload=true
Which looks promising, but I haven't read it carefully yet.
A: I think I have the answer. The spin was a concept I was referring in classical terms. I did not think about quantum mechanical concept of spin. So I think it would be wise to drop the usage of the word spin from spin angular momentum. When the sun rotates on its own axis it is angular momentum. The sun rotates on its axis once every 27 days. The spin I refer to is the sun rotating on its axis. So instead of using the term spin angular momentum we can simply say that the sun rotates on its axis and that simply refers to "Angular Momentum".
Also the sun orbits the centre of galaxy this is "Orbital Angular Momentum".  It takes the Sun 226 million years to completely orbit around the center of the Milky Way.
A: If by this you mean: does the Sun rotate as a rigid body the answer is certainly no, as you can read in this article.  BTW this article clearly refers to spin (in a non-quantum way of course) as internal angular momentum.
In addition some vortex flows have been observed; the internal and bulk rotational structure of the Sun is quite complicated (much to the delight of heliophysicists I might add).
A: Sun is a classical system, so we don't expect any quantum correlations. If any, the sun has a negligible spin in comparison with its angular momentum. The spin of the Sun is a sum of spins of all particles that constitute the sun, and this sum will be of the order of $\hbar$, because most of the spin will cancel. 
