Can Zhonya's Hourglass be reacted at any moment to choose which unit to save, including letting one unit die first or choosing between multiple units dying simultaneously?

No, you cannot react with Zhonya's Hourglass at "any moment" to choose which unit to save. Its timing depends on whether the deaths are simultaneous or sequential.

Simultaneous Deaths (Combat Damage) If multiple units would die at the same time (such as during the Combat Damage Step), you must have Zhonya's Hourglass face-up before damage is assigned.

The Choice: When multiple units die simultaneously, the controller of Zhonya's Hourglass chooses which one to save. The Timing: You cannot react during damage assignment or when damage is dealt. According to the rules, you must flip Zhonya's during the showdown while you still have priority. If it is hidden, you cannot wait to see the damage results before deciding to activate it. Sequential Deaths (Spells/Abilities) If units die at different times (for example, one unit dies to a spell on the chain, and another would die later to combat damage), Zhonya's Hourglass is mandatory.

It will automatically apply to the first unit that would die.