1. Suppose the second Chapter ability triggers twice, perhaps through clockspinning, and that I name the same card both times. Later, the third Chapter ability triggers and cast a spell with the chosen name. Do I draw two cards or four?
2. Same setup as question 1, except I choose different names. Later, the third Chapter ability triggers again. How many cards would I draw if I cast just a spell with the first name? Just a spell with the second name? Both?
Two.
There is no difference in meaning between "the thing" and "a thing". "The" merely indicates that we are referring to a thing that was just referenced. In other words, "the chosen name" means "a name chosen for the linked ability".
Every time you cast a spell, you ask yourself if the spell has a name chosen for the linked ability. If the answer is yes, you have cast a spell with the chosen name, and you must then ask yourself if it's the first time has happened this turn. If so, the ability is placed on the stack.
As such, the ability can resolve at most once per turn (unless it gets copied somehow), so you get to draw at most two cards per turn.
(The only other remotely possible alternative would be that the spell would have to have all of the chosen names, leading to zero cards drawn if different names were picked. But that is not the case.)
3. Does copying the second Chapter ability, such as with Strionic Resonator, have the same end result as 1 and 2?
It doesn't matter how you managed to get the second ability to resolve twice.
4. Suppose I have already named a card for Medomai's Prophecy's second ability, and then have Copy Enchantment enter as a copy of it. Does the new enchantment 'remember' the named card?
No, the chosen value is not a copyable attribute of Medomai's Prophecy.
706.2. When copying an object, the copy acquires the copiable values of the original object’s characteristics and, for an object on the stack, choices made when casting or activating it (mode, targets, the value of X, whether it was kicked, how it will affect multiple targets, and so on). The “copiable values” are the values derived from the text printed on the object (that text being name, mana cost, color indicator, card type, subtype, supertype, rules text, power, toughness, and/or loyalty), as modified by other copy effects, by its face-down status, and by “as . . . enters the battlefield” and “as . . . is turned face up” abilities that set power and toughness (and may also set additional characteristics). Other effects (including type-changing and text-changing effects), status, and counters are not copied.
706.7a If an ability causes a player to “choose a [value]” and a second, linked ability refers to that choice, the second ability is the only ability that can refer to that choice. An object doesn’t “remember” that choice and use it for other abilities it may copy later. If an object copies an ability that refers to a choice, but either (a) doesn’t copy that ability’s linked ability or (b) does copy the linked ability but no choice is made for it, then the choice is considered to be “undefined.” If an ability refers to an undefined choice, that part of the ability won’t do anything.
Example: Voice of All enters the battlefield and Unstable Shapeshifter copies it. Voice of All reads, in part, “As Voice of All enters the battlefield, choose a color.” and “Voice of All has protection from the chosen color.” Unstable Shapeshifter never had a chance for a color to be chosen for it, because it didn’t enter the battlefield as a Voice of All card, so the protection ability doesn’t protect it from anything at all.
Example: A Vesuvan Doppelganger enters the battlefield as a copy of Voice of All, and the Doppelganger’s controller chooses blue. Later, the Doppelganger copies Quirion Elves, which has the ability, “{T}: Add one mana of the chosen color.” Even though a color was chosen for the Doppelganger, it wasn’t chosen for the ability linked to the mana ability copied from the Elves. If that mana ability of the Doppelganger is activated, it will not produce mana.
The first example given by 706.7a is pertinent here.
[Partial credit to @GendoIkari for the answer to this fourth question.]
