Hi there Ashley! Thanks very much for taking the time to ask for clarifications! 
I'll do my best to provide a concise answer to each point:

I noticed that the fighter's "Open Doors" skill has a ≤3 symbol instead of a percentage. What does this mean? None of the other skills are listed that way. Is it that you roll a d20, and try to get less than or equal to three?

Opening Stuck Doors is dealt with in Part Five: The Dungeon:
"Sometimes you will come across a door that, while not barred or locked, has become stiff over the centuries of neglect. In such a case, characters should roll 1d6 to determine if they are able to unstick the door. On a roll of 1, the door becomes unstuck and can be opened normally."

So when making this check, the Fighter has a much greater chance - in the example give, (≤ 3) the Fighter can unstick a door on a roll of 3 or less on a d6. 

Wizards can only prepare an instance of a spell once per day. However, they are the only class that lists this requirement. Can other classes prepare multiple casts of a spell per day, or are Wizards just nerfed in that regard (and others?)

Wizards are indeed the only class this limitation is applied to. The design reason is to make arcane magic feel strange and alien compared to divine magic or the profound magical talents of the Elven Mystic. The spells leave the wizard's head when they are cast and need to be re-memorised like in Jack Vance's Dying Earth books. It does not effect the wizard too strongly in play but feel free to disregard the rule to suit your liking.

The Thief's Backstab skill says it is performed with "Combat Advantage." Is this just the same as regular advantage, that being +1d4 to the roll? Or is this something special? It's the only time I can find "Combat Advantage" listed in the book.

It is indeed the same as regular advantage. This wording is most likely a relic from an older draft that didn't get updated; I will change this for a future update!

The Thief's Assassinate skill. Both it and Backstab say the same thing, that they require an enemy to be unaware of your presence. Assassinate specifically calls out a Move Silently or Hide In Shadow check. What determines when a Backstab happens, and when an Assassinate happens? Or are they one and the same, and Backstabbing only happens when the opponent rolls a successful Save VS Death against Assassinate?

Assassinate is intended for use almost entirely outside of initiative. That is to say, on a sleeping opponent, or when a successful move silently check is made to sneak up on a sentry, or a successful hide in shadow check is made to ambush a patrolling guard. The purpose of the backstab damage occurring on a successful save is to simulate bad luck ruining an otherwise excellent strike. You might not kill your foe but you will wound them - then it might be customary for initiative to be rolled. 

Backstab is intended to allow for powerful strike while initiative is in play. Your thief might be invisible, or already hiding, or otherwise unseen, or it even might be that your foe is so engaged with another fighter that the referee might decide to grant you a backstab. Ultimately it is left fairly loose and open to interpretation by design by my proclivity it to allow both assassinate and backstab more than you disallow them - Thieves do not get much in the way of combat abilities or ways to slay large foes and this grants them exciting ways in which to do both! 

Is it necessary to puchase a saddle or saddlebags for a donkey/mule/other animal with a carry capacity? I don't see any listed, so it seems saddles and such would be included, but I want to be sure.

It is intended that saddlebags and barding are included with any purchase of an animal with a carry capacity.

Regarding the Usage Die: Do you roll for use on the first time an item with such a die is used? It says you get a free use, so I assume not, but I am checking to make sure.

When you use any item that has a usage die, you roll the usage die. The question the usage die resolves is 'was that my last torch/ration/flask of oil' what-have-you. Therefore, if you roll to light a torch, even if it is your first time doing so, you roll the dice. If the dice comes up one, you still use the torch you lit, but you have no more torches lit. This is the free use referred to. 
Remember that rolling a 1 on the first use doesn't mean your character was silly and only bought one torch/ration etc; it also represents accidental damage or spoilage!

Regarding Charisma and Monster Reactions. It says that people with exceptionally high or low charisma should be given bonuses "in line" with their Charisma to the roll. There's even a magic item that makes these bonuses the same as if you had 18 Charisma (Armor of Command.) But I cannot find the part of the book that details exactly WHAT bonus should be given. Is it the same as their maximum hirelings allowed? Is it the same as a To-Hit with weapons, but using Charisma instead? I'm not certain what bonus should be given

Although not stated, all Ability Score adjustments scale the same for each ability score: 9-12 = No Adjustment, 13-15 =+1, and so forth. That is to say you are right in that the bonus for a charismatic character applied to Monster Reactions would be equal to a To-Hit bonus provided by Strength. 

I hope this helps! Feel free to ask any follow-up questions. 
Happy Dungeoneering!