I'll give these a go
1. Explain to me how clerics choose spells and how wizards choose spells. I have walked around the rules for hours now trying to find concrete rules about how these two classes work and had to simply allow the cleric in our game to have access to all her level 1 spells and the wizard to choose only three as the example showed in the example of character creation.
You are correct, the Cleric has the entire range of Cleric spells to select from, while the Wizard starts with all the cantrips and three first-level spells in their spellbook.
The details are in the starting moves for those classes, on the
character sheets. The Cleric has the Commune move (p. 93 of the rulebook), while the Wizard has the Spellbook and Prepare Spells moves (p. 146). The Wizard's Spellbook move says
You have mastered several spells and inscribed them in your spellbook. You start out with three first level spells in your spellbook as well as the cantrips. Whenever you gain a level, you add a new spell of your level or lower to your spellbook.
So, that's the three first-level spells for the Wizard you were asking about, at least at first level.
The Commune move and the Prepare Spells moves are very similar to one another, allowing each character to prepare a selection of spells whose total levels don't exceed the character's level plus one.
I suppose the Cleric's move doesn't explicitly say they have the entire Cleric spell list to draw from, but that's implied because it doesn't outline any restriction.
The Wizard's Prepare Spells move, on the other hand,
does explicitly say they prepare spells
from their spellbook, so that (in conjunction with the details in the Spellbook move) is where the Wizard's limit comes from.
2. The Druid's shapeshifting move I understand, but the player was slightly confused as to why they can only hold the shape for only 1 or 3 moves. She felt extremely limited and was not satisfied with her character after that. I'm also wondering if it's possible to simply increase/decrease the druid's damage die depending on what kind of animal they turn into?
I think that for an appropriately dangerous animal form, having a move that adds extra damage to an attack is a perfectly reasonable move they could spend hold on.
Note that a "move" is not the same thing as a "turn". Your Druid could shift into a bird and fly around for hours of game time... that might cost 1 hold (if they were attacked and used a move like "escape to the skies"), or might not even cost hold (say, just to travel along above the rest of the party without doing anything in particular). The move says they have the animal's innate abilities and weaknesses, so it's only if they're doing something special that it costs hold.
Also, there's nothing that says they can't roll to get more hold while still in their animal form... it just exposes them to the risk of GM moves if they fail.
I've ruled in my game that their hold is expended when they
complete the move, so they don't revert to human form the instant they "escape to the skies" with their last point of hold, they can carry on for a bit until they actually land or stop trying to escape.
For some good examples of Druid moves in play, I've enjoyed reading a Dungeon World AP on Penny Arcade. Here are some examples:
1a- What actually happens, or is supposed to happen when something forces your alignment to change?
I'm not aware of anything that forces you to change alignment, but changing your alignment is discussed on pp. 34-35 of the rules. Mechanically, it simply changes what behaviour you earn XP for each session. In the fiction, changing alignment might be a big deal like an epiphany, or a slow slide from one alignment to another.
Note that you can choose a different alignment move for the same alignment, if the character has simply changed priority rather than changed fundamental morals.
2a- What's the deal with spell forgetting? It seems kinda.. broken, how a wizard can be rendered completely useless if their rolls fail. Hell even with a partial success you can make them forget spells, so what gives? D8
You or he must not have played much OD&D, where a Wizard starts with one spell that they can cast once a day :) In Dungeon World, they can cast the two first-level spells they prepared and all their cantrips multiple times each day, depending on the dice and their choices.
Note that there's nothing that says the Wizard in Dungeon World forgets their spell on a failed Cast a Spell roll. The GM can certainly do that as a hard move ("use up their resources"), but there are plenty of other moves he might make which don't result in the spell being forgotten.
On a partial success, the player can
choose to forget a spell, but if they don't like that option they can simply take one of the other two options instead. They might end up with reality so disturbed that they can't cast anything any more anyway, or dying from too much unwelcome attention, but that's their choice. Balancing those three options is part of the fun of playing a spellcaster.
3a - Is it possible to make a character that doesn't fit the stereotypical convention of its class. ie. can a wizard actually be weak in wizardry but excell in other areas and only use their title of wizard as misdirection to sleight other characters?
I'm not sure, but my feeling is not really, no. You could always assign your stat points such that your main class abilities aren't rolled with much bonus (or even with a penalty), but you wouldn't particularly be excelling in other areas.
There are more options at higher levels. Some classes (but not the Wizard) have cross-class advanced moves, allowing them to take moves from other classes when they've advanced a bit. There are also Compendium Classes, which give alternate advanced moves to select from. But basically, the game is built to emulate the OD&D classes, which didn't have a lot of scope for significant mechanical variation within a class.
You'd probably be better off playing (say) the Thief going around dressed as a Wizard, claiming that you're saving up your magic for things much more important than this, and possibly setting up tricks and judicious uses of Goldenroot to "prove" your magical abilities.