What's the right test case to find which method call is faster between execute and getElementsByTagName?

Answer 1

The right test case is to build two versions of your application, one of which uses each alternative but which are otherwise identical, and profile them in the typical situations you're trying to improve performance in.

Any other test would be potentially misleading as it's not measuring what you actually care about, which is application performance.

Alternatively, if you're not looking at this from the point of view of top-down application performance, then you should stop optimising prematurely and simply write the most understandable code right now (and optimise if/when you can measure that this block of code is making your app/library unacceptably sluggish).