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.
Emissary of Redemption is tracked by creature
The benefit only stops working on each creature that the paladin attacks, casts a spell on or deals damage to (emphasis mine):
If you attack a creature, cast a spell on it, or deal damage to it by any means but this feature, neither benefit works against that creature until you finish a long rest.
This means that casting beneficial spells on allies simply makes only them to no longer be affected by Emissary of Redemption. Unless you are worried that these allies might deal damage to you for some reason this should not be a problem.
Best Answer
Yes, if there are spell slots available to cast them with.
The Oath spells add choices, not spell slots, to the Paladin's magical arsenal.
To answer by way of illustration
Paladin, Oath of Vengeance, 5th level, Charisma 16 (+3 mod).
How many spell slots does the Paladin have to expend for this adventure day?
4 1st level spells, 2 2d level spells (p. 83, PHB, Spell Slots per level)
How many prepared spells does the Paladin have to choose from when deciding that he needs to cast a spell?
Preparing and casting Spells (p. 84 PHB)
Note that there are more spells available than you can prepare on a given day. How many total spells can you prepare?
Before we consider Oath Spells, you can prepare 3 + 5/2 = 5 spells total.
OK, pick a total of 5 spells from the list on page 208. Let's pick: 3 firsts and 2 seconds for simplicity
Bless, Command, Cure Wounds (1)
Magic Weapon, Lesser Restoration (2)
The Paladin can cast 4 first level spells, and 2 second level spells before running out of slots, and cannot cast any spell that was not prepared. (Such as Wrathful Smite, since that spell was not prepared). The paladin can instead cast 6 first level spells, by using a two second level slots to cast, for example, Cure Wounds cast at second level twice on grievously wounded comrades.
How do Oath Spells Figure into this?
Oath Spells increase the number of choices, not the number of spell slots. Add the following prepared spells for free to your prepared list.
Bane, Hunter's Mark
Hold Person, Misty step. (P. 88 PHB)
The paladin has the same 4 first level slots, and the same 2 second level slots, but can now choose from additional spells that were otherwise not prepared from the list: these are always prepared due to the oath chosen.
The choices, now, when choosing to cast a spell and having a slot to expend, are:
Bless, Command, Cure Wounds (1) Bane, Hunter's Mark
Magic Weapon, Lesser Restoratino (2) Hold Person, Misty step
Whichever spell is chosen, a spell slot is still expended when the spell is cast.
When the paladin gets to sixth level, he can prepare six spells (+3 + 6/2 = 6) from the list, and still has the Oath Spells prepared "for free" even though the paladin must always a slot to cast any spell.