The first part of your second question permits a rough calculation.
The second question ( i ): For humans the skin has been defined as ending at the hypoderm, which is 2-3mm deep. The average human skin has been estimated to cover an area of 1.5-2 m$^2.$ We can use 2.5 mm and 1.7 m$^2$, respectively. This gives a volume of 0.00425 m$^3$ or 4250 mL in fluid volume. The information for this calculation is at the Wiki site on human skin.
One estimate of the average total volume of fluid in a human body is 40L. Of that, 5L is blood. So roughly 1/8 of the body's fluid is blood. To a good approximation, the body is about 60% water by weight. This is presumably a little less than the percentage of fluids generally but since these are rough numbers we can use it. So 60% of the skin is fluid/water, and 1/8$th$ of this is blood.
Of the 4.250 L of skin, 60% is fluid, of which 1/8 $^{th}$ is blood, so $(4.25) \cdot (0.60) \cdot \frac{1}{8} = 0.3188$ liters of blood at any given moment in the skin, subject to considerable variation no doubt. This sounds a little low but the definition of skin sort of forces this--it is only skin-deep.
One source gives an estimate of the energy expended to pump blood in a normal adult heart as 0.5-1.0 Joule per beat which at 70 beats/minute amounts to 100.8 kJ/day using 1 Joule.
So the heart is handling about 5 L of blood of which 0.3188 L is blood in skin, so very roughly we could estimate that $0.3188/5 = 0.0638$ or 6.38% of the work above goes toward supplying skin, i.e. 6.43 kJ/day of energy.
As a sanity check, if someone burns 1200 [nutritional] calories a day, which is about 5 MJ, we can see that keeping skin supplied with blood represents a small fraction of daily energy expenditure.
The title question: Like you, I found no study on point but that does not mean one does not exist. I can't see why anyone would do/fund such a study as the answer is unlikely to inform further work, answer any current questions, or solve any problems.
The last part of your second question (ii) Transporting blood to the capillaries and circulating the blood through capillaries are aspects of the same job. Dividing the work of the heart into skin/non-skin was already artificial. To further divide the work into segments this way requires some motivation because any answer would be a bad approximation.
