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.
No, you can't cast it using a spell slot.
Spells you can cast because of your race aren't spells you know, and thus can't be spells you prepare. Per the PHB (p. 201, "Known and Prepared Spells"):
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind ...
... and it goes on to describe some of the class-specific variations of knowing and preparing spells. But in any case, having a feature from your race or class that allows you to cast a spell isn't the same as knowing the spell.
This is spelled out clearly in the multiclassing rules for Pact Magic quoted in the question (PHB p. 164; emphasis mine).
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
Likewise, the Drow Magic trait clearly distinguishes the known cantrip from the merely castable other spells (PHB, p. 24; bold for emphasis mine):
You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.
If the intent of the rules was that Drow also knew the non-cantrip spells, it would say so.
Best Answer
I'd say that the feat grants a Magical Training spell list, not a Sor/Wiz spell list.
When you take the feat, you get to pick 2-3 spells from the Sor/Wiz spell list, and those spells make up your spell list. You need to be an actual Sor/Wiz to gain their spell list.
Likewise, you're only considered a Sor/Wiz with regard to those selected spells, and for nothing else. So I'd say that you don't get to use items that require Sor/Wiz as the classes, but you still pass as an Arcane Spellcaster because those selected spells are arcane.
This is why the feat can pass the Dweomerkeeper's requirement (arcane spellcasting), but not exactly for the wands that you mentioned.