Who watches The Kagrumez Gauntlet from the balcony?

Has anyone looked up during the trials to see if there's anyone sitting there and if not, what is the role of the place?

I have not actually checked, but based on my experience in other Dwemer ruins, I'd be completely astonished if this were the case. Numerous NPCs and in-universe books repeatedly emphasize that the Dwemer vanished long before the events of Skyrim, and we never encounter any living Dwemer at any point in the game. I suppose it's theoretically possible that a member of one of the extant races of Tamriel might be up there, for some reason, but I'm not aware of any plot points which relate to such a thing. Regardless, the Dwemer probably didn't intend those chairs to be used by anyone other than themselves.

Did the Dwemer use it as an arena to test their subjects and to what end?

Similar chairs overlooking torture devices, arenas, and related paraphernalia can be found in Blackreach (in multiple places) and Bthardamz (the Lower District), and probably in other places that I'm forgetting. Based on some of the lore in Dawnguard (particularly their treatment of the Snow Elves), it seems like the Dwemer were not very nice people, in general, so it seems plausible to me that these devices were used on living subjects of some kind. Much of the lore emphasizes the Dwemer's focus on reason and logic, so I imagine that some form of scientific experimentation is the most likely explanation for these structures.

For more specific information, I would suggest consulting UESP's article on the Dwemer, and some of the in-universe books it links to.