[RPG] What happens when a Paladin falls from grace

Sword of the Arcane Order: Your spells are Wizard spells, not Paladin spells. They are arcane, not divine. This is because there is no specification that they count as Paladin spells. They may be consuming your Paladin/Ranger spell slots, but they still are Wizard spells, and when cast, they treat you as a Wizard with a caster level equal to your Paladin/Ranger/Wizard levels, added together. Take note that these are arcane spells, and are subject to Arcane Spell Failure. :(

Battle Blessing: This applies to Paladin spells, unfortunately, and have no bearing on the spells prepared via Sword of the Arcane Order.

Winter's Champion: The specified spells from the Cold and Winter domains are now in your Paladin spell list, and you may prepare them accordingly as Paladin spells (now subject to Battle Blessing). These spells are not automatically prepared, however.

Serenity: Nice choice for a Wisdom-focused Paladin. ;)

Please note that Sword of the Arcane Order does not automatically grant the Wizard spell list. Sword of the Arcane Order allows you to prepare Wizard spells, but you need a spellbook to base your preparation on. Essentially, your choice of Wizard spells is restricted to those in the spellbook available to you.

Also, if you don't have an arcane class that prepares a spellbook, you will always be treated as borrowing a spellbook (even if you actually own the spellbook). This means that you will be required to make Spellcraft checks in preparing each spell via Sword of the Arcane Order.

Revision Due to Comments

If I took the Magical Training feat, I would gain a spellbook, and three 0-level spells. With Sword of the Arcane Order, I would not be restricted from "borrowing from someone else's spellbook," correct?

Please be reminded that Magical Training can only be taken at 1st level. That said, unfortunately, the Magical Training feat restricts your learning to those three 0-level spells, and the Paladin has no 0-level spell slots, so you won't be able to use those with Sword of the Arcane Order.

Also, Magical Training does not grant the ability to prepare/write any other spell in the spellbook. It restricts you to those three 0-level spells. You will really need at least a 1-level dip into the Wizard class to gain the spellbook preparation/writing ability. Yes, you do own the spellbook, but to copy other spells into it requires the special ability of the Wizard, and this ability is not granted by the Magical Training feat. Even if another Wizard writes the spell into your spellbook, you will still need to decipher it and prepare it like a borrowed spell because you didn't write it yourself.

Prestige Classes are just like any other Classes, except where specific rule says something else.

From OGL content*:

Prestige classes offer a new form of multiclassing. Unlike the basic classes, characters must meet Requirements before they can take their first level of a prestige class. The rules for level advancement apply to this system, meaning the first step of advancement is always choosing a class. If a character does not meet the Requirements for a prestige class before that first step, that character cannot take the first level of that prestige class. Taking a prestige class does not incur the experience point penalties normally associated with multiclassing.

Emphasis mine. So if Paladin takes a levels if PrC, he is multiclassing all right.

Now, your player is wrong. Paladin can take any PrC he meets requirements for. The only issue is - he cannot ever take a level of Paladin again, unless PrC says he can. If prestige class has a rule that says that character can get levels of Paladin later, then she can, because specific beats general.

Example of such class (courtesy of Charlie) is Knight of the Chalice (Complete Warrior, p. 53):

Multiclass Note: A paladin who becomes a knight of the Chalice may continue advancing as a paladin.

So this class is an explicit exception. Some classes in Defenders of the Faith are exceptions, too. Probably few more scattered across sourcebooks. And that's all to it.

* Note that part about no XP penalty never made it to 3.5 DMG. It was in 3.5 OGL SRD, in 3.0 DMG and in 3.5 Complete Warrior. And 3.5 FAQ but, surprisingly, not errata. And bit different wording in each place. In this answer, I took wording from SRD, because that's where OP got Paladin class from, so this makes most sense to me.