Certain abilities require a free hand to perform: picking up an item or drawing a weapon, casting a spell with a somatic component, using Gloves of Missile Snaring, etc.
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If you make a two-handed attack with a Versatile weapon, for how long are both hands occupied? Only during the instant of the attack? Until the beginning of your next (one-handed) attack? Until you explicitly take a free action to change your grip during your turn?
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Can you drop a weapon (or other object) as part of a reaction to instantly have a free hand? (For example, to use the aforementioned Gloves of Missile Snaring)
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Can you change your grip on a versatile weapon as part of a reaction to instantly have a free hand?
Best Answer
The important rules, before we begin:
This makes the answers to your 2nd and 3rd questions easy. You cannot perform either flourishes or Object Interaction when it is not your turn, and therefore cannot do either as part of a Reaction (unless your Reaction somehow happens during your turn). The answer to your first question is a little harder. As previous discussions here and here show, the official rules on what requires you to use your once-per-turn Object Interaction are open to interpretation. Some people consider opening your hand to count as one of your freer-than-free flourishes while others consider it to be an Object Interaction. So you're getting two answers.
Answer 1: Opening Your Hand is Freer-Than-Free
This position takes the stance that opening your hand is freer-than-free. Think of it as Object Uninteraction. This position happens to be well supported by developer tweets, which have variously stated exactly that, as well as even further oddities such as Crawford stating that Clerics can cast somatic spells while holding a weapon and shield - without even needing the War Caster feat. This position also happens to be quite popular, though we should be wary that this popularity is due to players who'd love to have the benefit of a free hand regardless of their other choices. On the other hand, consider: if I were to use my Object Interaction to open a door, and in the process of opening my hand to turn the door knob, a sword happened to fall out of my hand. Too bad for the sword.
The implication is therefore that you can drop an item, or shift from a two-handed-grip to a one-handed-grip - entirely for free so long as it occurs on your turn. Note that the rules are quite strict that whether something is a flourish or an Object Interaction, it happens on your turn, and in the case of Object Interaction, either during your Move or Action.
Thus: You can release your grip of one or both hands at any time during your turn without spending your Object Interaction. Restoring your grip requires that you use your Object Interaction.
Answer 2: Opening Your Hand is an Object Interaction
This position is not without merit, as allowing players to have a free hand effectively whenever they want strains game balance, and the idea of characters having free hands even when both are full strains credulity; however, this position is lacking in support from developer comments.
If we assume that opening your hand is an Object Interaction, then you can shift a versatile weapon from one hand to two hands, or from two hands to one hand, on your turn. But you can't do both.
Answer 3: Didn't You Say There'd Be Two Answers?
But Wait, There's More! Another possible interpretation could be that the user chooses an Object to interact with, and can continue doing so at different points in their turn.
This position would allow a player to shift from one hands to two, or from two hands to one, multiple times on any given turn.
Final Notes
Reminder that regardless of which position you take, whatever you end your turn with is what you're stuck with, RAW.
My personal feelings? Well, imagine you're reaching out to turn a door knob or performing some arcane gestures, and a sword happens to fall out of your hand. Too bad for the sword.