Can a particle have an “actual” spin? I know that a particle can have an orbital momentum and a spin which is intrinsic to the particle and doesn’t really have to do with the particle spinning. But can particles have an additional momentum due to its “actual” spin? Can a particle spin in the first place? And if so, what are the consequences of this motion?
 A: Here is some perspective which may help. 
We know that if a macroscopically-sized and charged object is physically spinning, it will exhibit a magnetic moment. Experiments demonstrate that the electron actually possesses a magnetic moment, but other experiments demonstrate that as near as can be determined, the electron is a point that has no diameter. So, if it has a magnetic moment but has no diameter, then what exactly is doing the "spinning" in the case of an electron?
Years ago, the physicists studying this did a calculation in which they plugged into the equation for the magnetic moment the known value for that of an electron, its charge, and a reasonable guess for its diameter (based on an experimental upper bound) and discovered that to produce the measured value for its magnetic moment, the electron would have to be spinning so fast that at its "equator", it would be moving much faster than the speed of light. 
This meant that the classical concept of "spin" for big objects fundamentally did not hold for something like an electron, and that it was not useful to imagine that a point particle was physically "spinning" around its central axis. 
So there you are: the electron presents us with a magnetic moment, suggesting it is revolving- while at the same time it has no diameter, and even if it did, there's no way for it to generate that moment without violating special relativity.  
Welcome to the world of quantum mechanics!
A: 
My question is: can a particle actually spin? Like, classical spinning. 

Two answers: no and nooooooooo :). 
First off, we don't have accurate measurements of the size, if it actually has any, of an electron, for example. 
All of the properties we give elementary particles are chosen to make sense when scaled up to the classical world, but they are analogies, we really don't know what exists, if anything,  at the tiniest scales.
What you are asking is really a philosophical question, in my opinion, because physically/experimentally  we can't measure below a certain scale. 
So I can't say for sure that there isn't a minute "thing" spinning, despite my silly comment at the top, but what an electron "is" and how it behaves. seems much more likely to have a more subtle answer than anything remotely classical.
Virtual and Real Particles is a good read for how a particle is described in quantum field theory.

If we aren’t sure that particles are of finite size, then how could scientist show that the particle wasn’t actually spinning? Even at really high rotations (but still slower than the speed of the light) wouldn’t the magnetic moment be measurably high? 

We can't be absolutely sure that it isn't spinning, but classical spinning wouldn't explain the behaviour of an electron, in any event.
A: All we can say with certainty is that we do not have a mechanical model of the electron. Its size appears smaller than can be detected and then there is the half integer value. This makes it hard to represent it by a small spinning object. If we do, we need it to spin so fast that its outer parts move at near light speed, with a large gamma value. The point is that relativity is not the issue. Dynamically electron spin is like any, quantized, angular momentum.
