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.
Yes, they stack perfectly well.
Neither ability requires an action to use, so as long as you hit with a melee weapon attack that happens to use your pact weapon you can use both. The only exception is that Eldritch Smite can only be used once per turn, while Divine Smite can be used as many times as you have melee weapon attacks.
If the intent was to use only one at a time it would be called out in the text or require a bonus action or something else that is clearly not the case here.
Best Answer
No
Mechanically speaking, a Divine Smite is a Paladin class ability. Meanwhile, a spell is anything listed in PHB pages 207-211, which shows the spell list for all classes. So even though Divine Smites use spell slots, they are not spells.
Not all spells consume spell slots
From PHB 201:
Therefore, spells and spell slots do not have to be tied together, though they usually are. Spells can be cast without spell slots. Just because something consumes a spell slot doesn't mean it has to be a spell.
Antimagic Field does affect it
While a Divine Smite is not a spell, it is magical. "Magical things" is a larger set that contains all spells.
Antimagic Field specifies:
It specifically states:
As Divine Smite is a magical effect, it is nullified in an Antimagic Field.
It does work in an area of silence
Divine Smite is not a spell and requires no verbal spell components. All it requires is you hit with a weapon and you expend a spell slot.