There‘s a great video on that topic by Matt Parker (Standupmaths) which I‘d wholeheartedly recommend.
TLDW: No, terrain elevation isn‘t generally taken into account when calculating a country‘s surface area, though the actual method differs slightly from country to country. Switzerland would be around 7% bigger, if its mountains were to be taken into account.
I‘m personally very happy with kagi when it comes to features and, most importantly, the quality of search results. But, as they don‘t have ads, it’s pretty expensive to use.