Is it more reasonable to use stratification to separate similar compounds rather than using temperature? In a simple reaction between butane and chlorine, it is more likely for the secondary carbons of the butane to lose a hydrogen and gain a chlorine. The carbocation is a more stable intermediate in that situation compared to a primary carbon losing the hydrogen. So if the desired synthesis is 1-chlorobutane as opposed to 2-chlorobutane then it seems reasonable that a chemist might use stratification. The reason that I believe it is less desirable to use temperature as a method of separation is because that, at least in this example, both compounds will move between the gas and liquid phase randomly providing the possibility of still having mixed samples. Is this true, and will stratification work more reliably if purification is the desired result?

Furthermore, I never really learned why mono-chlorinated products form. Will there not also be compounds that have been chlorinated twice? Or will, over time due to entropy, all compounds be mono-chlorinated assuming that the reagents are in a ratio of 1 chlorine atom per molecule of butane?
The answer to your second question:

The halogenation of saturated hydrocarbons occurs by free radical substitution in the presence of UV light or heat and not by formation of carbocations. There's a very clear representation of the reaction mechanism on this Wikipedia page.

Since free radical intermediates are involved, this is a chain reaction and so what you said is correct, a large number of substituted products can be formed depending on the reaction conditions. 

For example: If you consider the simple reaction between Methane and Chlorine in the presence of sunlight, Chloromethane, Dichloromethane, Chloroform, Carbon tetrachloride and a small amount of Ethane (by combination of Methyl radicals) are formed depending on the amount of chlorine available.

All these products will be formed even if the reactants are in a 1:1 ratio but they will be present in low concentration since there is less chlorine available.

In the case of higher alkanes like Butane and Pentane, the secondary carbons are substituted faster since the secondary free radical intermediates are more stable (reasons are the same as that for secondary carbocations).