As previous discussions [here] 1 and here show, the official rules on what requires you to use your "Object Interaction" are open to interpretation. That said, the majority so far seem to be taking the side that you don't need to use your "Object Interaction" to open one or both of your hands - though you would need to use it if you wanted to pick back up the thing you just dropped, and you would need to use it to restore your grip on a two-handed or versatile item. Let's refresh ourselves on the rules once more:
Your turn can include a variety of flourishes that require neither your action nor your move. ...
You can also interact with one object or feature of the environment for free, during either your Move or Action. ...
If you want to interact with a second object, you need to use your action. ...
Player's Handbook, pg. 190
We see that your turn can include a variety of flourishes, independent of interacting with objects. As stated above, the popular answers on this site indicate that many people believe that opening your hand to drop something counts as one of these flourishes - but this is ultimately the purview of the game's Dungeon Master. That said, all of the rules above are very strict about applying to your turn, and in the case of Object Interaction, either during your Move or Action.
So to answer your questions:
You can release your grip of one or both hands at any time during your turn. Restoring your grip requires that you use your free Object Interaction.
(and 3.) 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).
To be perfectly clear, my answer to #1 is dependent on majority opinion, and any given DM would be well within their right to rule differently. My answers to #2 & #3 should be rules-as-written.