It would be the sum of the areas of each surface separately.
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No. In fact, if they retain their combined volume, their surface area would increase.
A surface area would be vital for determining volume
No, they are two different things. Area encompasses the total object while Surface Area only encompasses what it implies. For instance, if you had a box, the Area would be the total size of the box, while Surface Area would only indicate the size of the top of the box. no. area is Lenth times width and surface area is what the area of the whole object.
It would help if the question was less obscure. What do you mean by "work"? How the surface area affects chemical processes (for example the surface area of catalysts), or diffusion, or surface areas and friction?
A perfect dome (1/2 of a sphere) has a surface area of 2piR2. This is because a sphere's surface area is 4piR2, and dividing that by 2 would simplify to 2piR2.