You and @KRyan are correct.

• all multipliers work alike (crit, lance, brace, whatever);
• everything is included in the multiplier except bonus dice
• multipliers are additive not multiplicative, see Multiplying Damage, Core Rulebook 179

It doesn't say anything more about it because there's nothing more to say, the rule is simple and all-encompassing.

Your math is slightly off however. For a tenth level paladin with STR 22, Spirited Charge (req: Mounted Combat, Ride-By Attack), Power Attack...

• BAB +10
• Lance: 1d8 damage
• Magic enhancement: +2 to hit/+2 to damage
• Strength: +6 to hit/+9 to damage assuming you're two-handing the lance
• Power Attack: -3 to hit/+9 to damage
• Base Melee Attack: +15 to hit, 1d8+20 damage
• Charging: +2 to hit (-2 to AC)
• Smite: +10 damage vs evil, +20 vs "super-evil"
• Mounted and Spirited Charging: +17 to hit, 3d8+60 damage
• Smiting Too: 3d8+90 damage vs evil, 3d8+120 vs super-evil
• On Crit: 5d8+100 vs normal, 5d8+150 vs evil, 5d8+200 vs super-evil
• The aristocrats.

On a crit, you can one-shot a Hezrou (CR 11). It'll live through the first non-crit shot, but you can get it with the next normal blow (assuming you make your save against the blasphemy...). Yes, that's how it's supposed to work. Yes, it's pretty hardcore. You can only do it 4x/day and need to set up a mounted charge and to crit, though. But many super-optimized builds are built around charging for this reason.

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.