Why would you say "je vais me faire l'avocat du diable" instead of "je vais faire l'avocat du diable"?
"Se faire" can have two meanings.
The first one is used in a sentence where someone does something to the subject.
In this case, Paul is the subject of the sentence, but he is the one who is recieving the beating. In can be compared to an English present continuous passive sentence (how twisted!), the passive of "John is beating Paul".
But there is a particular meaning, when the subject of the sentence is the one who does the action.
In this case, "se faire" means "to make oneself" or "to turn oneself into". Literally or figuratively. You can say "La chenille se fait papillon" (The caterpillar turns itself into a butterfly).
Back to your example:
Both are correct and used.
As the writing is similar, we usually don't see the difference in French, especially when speaking. But "I make myself the devil's advocate" implies that you are trying to look like one, even if you are not, pretty much like "I make myself big" of "I make myself strong".
Don't confuse that with "Je vais me faire un café" (I will make myself a coffee), which is a third meaning that I shall not discuss here, for it has a completely different context.
Les deux peuvent se dire mais le rôle du locuteur est différent à chaque fois.
We can say both, but they each have a different meaning.
Also be careful, because this expression is very ambiguous... Usually "se faire quelqu'un" means "do someone" (to have a sexual relationship with someone).
Here, it is not the case I guess, it means "to make yourself as", "to act as"