There seems to be some confusion here regarding language. In your first paragraph, you define the free monad generated by an endofunctor $R$, i.e. one first fixes $R$ then gets the monad. A good toy example for such $R$ would be $U\circ F$ where $F:X\to Y$ is the free functor into some category $Y$ which is well understood and $U:Y\to X$ is the forgetful functor. So the answer to your first question is yes, it can take this form and that's a nice case to play with. But in general $R$ need not take this form. Theorem 5.5 in the paper you cite gives a sufficient condition on an endofunctor $R$ so that free monad on $R$ exists. That condition has nothing to do with $R$ coming from an adjunction.
I can interpret your second question in a couple of ways. One way is as "when is $T$ of the form $U\circ F$?'' This is trivial; it's well-known that a monad $T$ is always of the form $U\circ F$ (see wikipedia), often for more than one choice of $U,F$. A better way to mean given interpret your question is "given a monad $T$, how do I determine if it's the free monad generated by some $R$ which takes the form $U\circ F$?" This seems appears to be a non-trivial problem. Theorem 5.4 in the paper gives one situation where you can determine that your monad is free and recover $R$, but this doesn't classify all such situations. Anyway, even in this nice case where you get $R$ in hand, there are plenty of times such $R$ could fail to be in the form $U\circ F$. For instance, this MO answer shows that any such $R$ would need to be a homotopy equivalence on the nerve of $X$. It should not be too hard to construct an endofunctor for which this fails, since there are plenty of self-maps of simplicial complexes which are not homotopy equivalences.I like to think about monads as monoids in the category of endofunctors. I'd be interested to know which types of monoids have the form $U\circ F$. I wonder if there's an algebraic characterization
There seems to be some confusion here regarding language. In your first paragraph, you define the free monad generated by an endofunctor $R$, i.e. one first fixes $R$ then gets the monad. A good toy example for such $R$ would be $U\circ F$ where $F:X\to Y$ is the free functor into some category $Y$ which is well understood and $U:Y\to X$ is the forgetful functor. So the answer to your first question is yes, it can take this form and that's a nice case to play with. But in general $R$ need not take this form. Theorem 5.5 in the paper you cite gives a sufficient condition on an endofunctor $R$ so that free monad on $R$ exists. That condition has nothing to do with $R$ coming from an adjunction.
I interpret your second question to mean given a monad $T$, how do I determine if it's the free monad generated by some $R$ which takes the form $U\circ F$?" This seems to be a non-trivial problem. Theorem 5.4 in the paper gives one situation where you can determine that your monad is free and recover $R$, but this doesn't classify all such situations. Anyway, even in this nice case where you get $R$ in hand, there are plenty of times such $R$ could fail to be in the form $U\circ F$. For instance, this MO answer shows that any such $R$ would need to be a homotopy equivalence on the nerve of $X$. It should not be too hard to construct an endofunctor for which this fails, since there are plenty of self-maps of simplicial complexes which are not homotopy equivalences. I like to think about monads as monoids in the category of endofunctors. I'd be interested to know which types of monoids have the form $U\circ F$. I wonder if there's an algebraic characterization