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.
Update October 2020: No
As user @V2Blast mentions below, the Sage Advice Compendium page 13 explicitly rules "No" for this question. Page 13:
Can I make an attack with one weapon, then draw a second weapon with
my other hand and qualify to use two-weapon fighting?
To use the
two-weapon fighting bonus action (PH, 195), you must have both weapons
in hand when you make the first attack. If you’re instead fighting
with two or more weapons as part of the Extra Attack feature, the rule
for the two-weapon fighting bonus action doesn’t apply. The rule for
that bonus action applies only to itself, not to any other use of two
or more weapons in the game
Original Post
No, you cannot draw your dagger and attack with it while retaining your Duelist feature
But you can draw your dagger as part of an attack and then use it.
The core argument behind this answer is that you can only attack as a bonus action with your offhand weapon when you use your action to attack.
When you take the Attack action and attack with a light melee weapon
that you’re holding in one hand, you can use a bonus action to attack
with a different light melee weapon that you’re holding in the other
hand. (Basic Rules, p. 74 / PHB. p.195)
You don't have this bonus action otherwise.
You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus action. You otherwise don’t have a bonus action to take. (Basic Rules p. 69)
You can draw your weapon as part of an attack...
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack. (PHB p. 190)
...but that would remove your +2 bonus
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon. (Basic Rules, p. 25)
So, here is what you are trying to do.
You attack
You can use Two-Weapon Fighting to attack with a weapon that is already in your hand. You can subvert this by drawing your weapon as part of your aforementioned attack declaration.
Since you now have another weapon in your hand, you lose the Duelist feature of +2 damage.
If you forgo drawing your dagger, you are not allowed a second attack since the dagger is not in your hand during your attack action. It's either one or the other, you cannot combine two-weapon fighting and duelist in the same attack action. You can dual wield and drop your dagger as a free action before you attack with it to retain your Duelist bonus for the next turn (turn start, attack with no bonus, attack with dagger, turn end, turn start, drop dagger, attack with bonus)
If you need a logical reason for why this has to be, you can think of a duelist as sort of "fencing" with his body turned to the side, putting all his focus on the one weapon. Dual Wielding is less precise and doesn't allow your character to focus on the one attack, but spreads his focus out over two weapons.
One final note, you cannot use your Rapier for Two-weapon fighting because it does not have the light attribute, unless you have the Dual Wielder feat.
Best Answer
Jeremy Crawford is saying it's not an action to shift a weapon between one or two hands. So you could draw you dagger as part of a move or as part of your Attack action, after you fired the bow twice.
However, I do not know of any way to use a bonus action to attack in your example, as the situation doesn't qualify for dual wielding, nor the Crossbow Expert's condition for bonus attack.