How to identify if a chemical equation has a 'fractional order'?
It is not possible to determine the rate law of a reaction based on solely the balanced reaction. We must determine the rate law by experiment or by an established mechanism for the reaction.

What is meant by a mechanism is a series of elementary steps, a set of reactions whose linear combination yields the balanced reaction. Each elementary step represents a microscopic collision that happens during the process of the reaction, and these elementary steps obey the law of mass action, which allows us to write the rate law for each step in the reaction.

We then make a few approximations, perhaps guided by experimental observation, to obtain the rate law for the reaction. For example, we may conclude that one elementary step is rate-determining, so that the rate law for the reaction is governed by the rate law for that step; we may make a quasi-equilibrium approximation, which means that the species involved in an elementary step are equilibrated; or we may make a pseudo-steady-state assumption, which states that the net rate of production of an intermediate, presumed to be short-lived, is zero. 

How do we know that a proposed mechanism is correct? Well, we don't. All we can do is prove that a mechanism is incorrect, by establishing that it posits the wrong rate law as found from experiment; we cannot distinguish between different mechanisms that result in the correct rate law. Nevertheless, we can often obtain much indirect evidence to support a given rate law, and we can frequently make good guesses based on patterns from experimental observations.

For example, while I wouldn't be able to accurately guess the rate law for the reaction you've provided, I can at least say that it is likely to be of half-integer order in chlorine, because such reactions often follow a radical mechanism, which often lead to non-integral powers in rate laws.