Why does a frog's oxygen intake through the skin and lungs vary throughout the year?
Cutaneous respiration is based, as every type of respiration, on diffusion of gases from a higher concentration (in water or air) to a lower one (in capillaries). 

Citing from Johnson, Peter H. Raven, George B. (2002). "Respiration". Biology (6th ed.). Boston: McGraw-Hill. pp. 1053–1070:

In general, the rate of diffusion between two regions is governed by a relationship known as Fick’s Law of Diffusion:

R = D - A x ∆p/d


In this equation,


R = the rate of diffusion; the amount of oxygen or carbon dioxide diffusing per unit of time;
D = the diffusion constant;
A = the area over which diffusion takes place;
∆p = the difference in concentration (for gases, the difference in their partial pressures) between the interior of the organism and the external environment; and
d = the distance across which diffusion takes place.


As you can see, the rate of diffusion R can be increased:


By increasing A, the diffusion area;
By increasing ∆p, the difference in concentration; 
By decreasing d, the diffusion distance.


In cutaneous respiration, nr.2 is not really an option; yes, capillaries can be optimised to bring blood rich in carbon dioxide and poor in oxygen really close to skin surface (which brings us to nr.3), but it's a lot less efficient than, for instance, lungs that can exhale air filled with carbon dioxide and replace it with fresh air. While the intake of oxygen of the skin remains more or less the same even with different metabolic demand, lungs can quickly adjust the breathing rate depending on the metabolic demand of the moment.

So, the answer to the question


  why does the intake through the skin vary slightly while through the lungs dramatically?


is that skin cannot regulate its gas exchange rate, while lungs can. This brings us to your first question:


  Why does a frogs oxygen intake through the skin and lungs vary throughout the year?


The answer lies in the biological cycle of the frog throughout the year: during cold months, frogs hibernate. In winter, frogs slow dramatically their metabolism and they don't need so much oxygen; thus, in this season they are able to rely only on cutaneous respiration. In other seasons they are active, and they have to breathe actively through their lungs to supply their blood with the necessary quantity of oxygen.