Maxperson
Morkus from Orkus
Example 1: A barbarian, mechanically, who is a rural peasant who flies into a rage and fights with exceptional vigor on behalf of the lower class, and harbors a hatred of the nobility. I'd call this a minor refluff.
Fighting on behalf of the lower class and harboring a hatred of nobility doesn't have anything to do with any specific class. It can fit them all, so the only thing you have there is a rural peasant who flies into a rage. That doesn't really tell me enough. Does he fit the general theme of the Primal Instinct portion of the barbarian description? If no, I'd put that at moderate. If yes, it isn't even a refluff.
Example 2: A bard, mechanically, who is flavored as a mage with no musical skills at all. They have a keen interest in magical theory and see themselves as an unconventional wizard sussing out the secrets of magic from many traditions. Basically a bard who is flavored like a more traditional caster. The two are fairly close mechanically to begin with, so I'd call this a moderate refluff.
A PC can call himself whatever he wishes. If you want to call yourself a mage, you can do so. As for having no musical skills, even though the spellcasting section says that they shape reality with wishes and music, their abilities mention music and words as how they use those abilities. I would rate that a minor refluff, since all you are really doing is allowing oration to be how you cast spells, rather than it being music.
Example 3: A druid, mechanically, who is played in the style of Radagast, complete with a bunny sled. Radagast is more of a wizard who is attuned to nature than a druid, so I'd call this a moderate to high refluff.
Radagast has very little known about him other than he is close to nature. Tolkien called them all wizards, but I see no reason why he couldn't have been a druid, and the character in the movie could also have been a druid. All your description really changes is that this character is calling himself a wizard. If he plays like Radagast, then regardless of a sled, he's acting like a druid. This isn't even a refluff as far as I'm concerned.
Example 4: A monk, mechanically, who is a hobbit farmer who brawls and hits people with his hoe and utilizes old family fighting traditions ("Aunt Winnifred's Crotch Punch" for stunning strike, as an example). This departs pretty drastically from the monk flavor, so I'd call this a heavy refluff.
Yes. This is absolutely a major departure from the fluff. I would allow it in my game, though. I don't mind homebrew or house rules.
Example 5: A paladin, mechanically, who is flavored to be a mechanical assassin who unleashes surges of energy through his weapons in the form of energy blasts on a hit (to replace smites), and who swears no oath. I'd call this an extreme reflavor.
This is basically the creation of a new class and based on the new fluff of "mechanical assassin," would require more alteration before I would allow it. It doesn't make sense to me that his assassin would be able to lay on hands, cure diseases, be immune to disease, give bonuses to saves to others, etc. I would work with the player to try and come up with something, though. If we could come up with new class abilities that seem to be in the same power level ballpark as the ones that don't make sense, I would allow this.