How they determine a country's surface area?

"flat" like with a map on the larger scale.  On the property scale, the sides are typically "true" distances (straight line measurements between two points without regard to whether the line is truly horizontal or slightly out of plane) so may take into some account the impact of elevation change (if the change is constant across the length of the line).  The country area is often put together (calculated) by summing up the areas of many regions, and regions are often the result of totaling the declared surface areas of towns or whatever, which can themselves be a total of the sum of all property measurements, but are not always.  So, they do not try to account for vertical effects on true area, but some of the data do that, and some do not (not everywhere on earth has been surveyed properly).  It depends a lot on what you use for base figures.

Typically, the area values are a ballpark estimate anyway and not intended to be precise at the level that worrying about vertical distance impacts would require.  You are talking about effects that are down at the decimal percent (third to fifth significant digit details) so already lost in the lack of precision of the given number.  Most areas are only stated to 2 or 3 significant digits.  Any claims to be more precise are not valid.  They were not calculated using true measurements at the required level of precision to get there.  Especially when there are irregular coastlines involved.

I've never seen a country's area listed as "surface area", which is pretty difficult to measure. Do you measure every gully? how about the cities, do you measure the surface of every buildings?

Usual number is just "area" which assumes it is flat.