The important rules, before we begin:
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
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.
The DM is charged with making rulings on a huge variety of things that go on in the course of playing the game. You can make your case for why you think it should be a given way, and then await a ruling.
Once the ruling has been made at the table, the DM is right1.
During play, accept that and then press on as the other players wish to play for fun and are probably not there to watch an argument.
Once play is done for that session, you have reached a potential decision point.
Is this particular decision a deal breaker for you? Do you want to revisit it outside the time constraints of a gaming session, in a non-confrontational manner? If not, if mostly you are having fun, then roll with it.
In this case, you picked that cantrip with the expectation that you could do certain things, and have just found out that you can't. In a non-game time environment (or in a friendly email) present the PoV that your expectations were not met (by accident) and ask for another cantrip. Or, ask that she reconsider the ruling once you've explained your position. Your request must be unemotional and non-confrontational.
DMs do this for fun, not for pay.
Once you've re-stated your case, accept any follow-on ruling with good grace, pro or con. She does a lot of work to run this game for your group.
Peace between you two will benefit the whole table.
If this ruling is a deal breaker ... if this is a decision in a pattern of rulings that you find are consistently dashing your expectations against the rocks ... then you need to have a different (unemotional) dialogue with the DM:
Are you two playing the same game with the same expectations?
If you can't reconcile that, this table may not be a good fit for you.
Even if you don't always see eye-to-eye, there are some things that you can do. You can contribute some good faith effort to future decision points.
Without going through the eye-watering detail of the Same Page Tool, it's worth your while to look at spells that you are interested in adding to your spell book before you get to them. Likewise the spells you already have.
Do some homework, and a little forecasting. Try out some practice scenarios yourself. What would I do in X case? In Y case?
In an email or a conversation outside of a game session, present some of your ideas on how a spell might work beyond its obvious uses. A lot of DMs appreciate ideas raised and resolved before or after the gaming session where a non time-critical decision is achievable. (I sure did when I was running games during my early DMing years, and I really appreciate it now that I am running them again. My group has a lot of "how does this work?" and "I thought it worked like this" episodes).
You may be able to get her to "see it your way" a few more times when she's not under time pressure. Or not. However this dialogue plays out will inform your decision on whether this is a good table for you, or not.
Insofar as the interpersonal skills: the more non-confrontational you make your approach to her, the more likely you are to get her to see it your way on some issues, but you won't always get your way.
Why?
At the end of the day, the DM's rulings are the rule. She's got more than your fun to consider, she has the whole table's fun in her hands as DM.
Best wishes for continued fun at this table, or any other.
1 (PHB, p 6)
Ultimately, the Dungeon Master is the authority on the campaign and
its setting, even if the setting is a published world.
Best Answer
Trinkets can be wearable items, so Prestidigitation could create medical gloves
'Trinket' is not a specifically defined game term.
However, there is a list of 100 trinkets in the Player's Handbook from which it's possible to derive some parameters.
The following items are all included on this list:
From this it is safe to conclude that trinkets can be wearable items.
I therefore would allow Prestidigitation to create a pair of wearable medical gloves.
As per the spell description, they still wouldn't last beyond the end of your next turn, however, which probably isn't really enough time to make any use of having them.