For me in the US also, 'make' definitely seems much more natural. Though I probably wouldn't outright reject 'construct' (which they accepted Jan. 2018) or perhaps 'build' either, I generally prefer to use those words for bigger projects, such as houses or bridges. For the things that a craftsman, artisan, or hobbyist might make, including furniture, plastic models, and jewelry, I tend to use make, though there is undoubtedly some area of overlap. Even in the figurative area you can build bridges, construct a theory (i.e. the bigger things), but make a (small) personal guess.
The two answers "furniture" and "pieces of furniture" are given in in the drop-down list for the new word; I chose the second; and I was marked wrong. It is very frustrating to lose hearts on a series of these apparently capricious decisions, which run counter to the actual teaching in the lesson.
The problem is that "mobili" means furniture; it's uncountable in English, but not so in Italian, so to translate the Italian "mobile" you don't really have an equivalent. Duolingo chose to use "piece of furniture" for "mobile", but it isn't always consistent in its application.
Not true. I have used that phrase. In this context, make is the same as build. Make is just a general term that can be used for almost anything that produces an object. Build is correct as well. Build roughly means to put pieces of something together (not mixing together like making a cake), to construct something larger.