Very simple -
Your skills and Magic learned is from Before you were a Vampire. Doesn't matter if you become a Vampire after learning how to use "Blessed" Magic from some Holy Order, at the end of the day being a Vampire doesn't take away your skills or change what you know.
Summoning Daedra as a Sorcerer, working with the Fighter's Guild as a Werewolf, embracing the power of the Undead to aid you against Molag Bal - In the end, it all seems to me like a land torn apart by war and a "world" under siege by a Daedric Prince isn't exactly the right place for people to be picky whom they call ally - and ultimately the magic, powers, and skills that you can use as a character have less to do with any religion or affiliation than they do with just powers your character can grasp, understand, and utilize.
Does it honestly make sense for a Vampire to be able to use the "Dawnbreaker"? Yes. Is it risky for a Vampire to use "Dawnbreaker"? Well, yeah... Is that reflected in this game? No. Are there "Lore Friendly" explanations why a Vampire Can safely use it (Or any other powerful artifact/Power related to a Godly force that hates undead)? Yup, plenty.

At the end of the day, your Vampire is still fighting a war against Molag Bal, and is still the most likely to bring victory and stop his Schemes, as well as the plots of any others who come after him. These "Godly" forces make exceptions, make deals, make bargains, and play favorites.
Regardless of what you may think of the Aedra, Daedra, or Magna Ge - You must keep one thing in Mind. None of them are Good. At best, they are self-absorbed and distant forces of Power that are entirely absorbed in their own narrow aspects and views of the world, eternally selfish and fully focused on an "ends justifies the means" style of enforcing their Aspects as Law.
At worst, these God-like beings are malicious forces that only tolerate mortals as long as they can prove entertaining or useful.
You can not know the alien minds of these "Gods", nor can you hazard a guess at their motivations and opinions, even if they were to tell you to your face.

It makes perfect sense that a vampiric daedra-worshiping citizen-grinding sociopath could use Aedric Power as long as at the end of their blood-mad rampage, they still kick Molag in his balls and stop the Planemeld.
Let us never forget Pelinal "Kill-All-Elves" Whitestrake, a genocidal maniac who hates everything non-man, prone to bursts of insanity, and a man who slaughtered his way through a healthy chunk of the Khajiit population before realizing they weren't elves and loosing interest and wandering off, was full of "Divine" power.
Stop pretending "Templar" means anything "Good" or even slightly "Holy"

Honestly, the Dark Brotherhood are the most honest out of anyone - They worship a "God" who thinks creation was a huge mistake and wants to end it and reclaim all souls. At least they're not trying to cover the bloodlust, they just embrace it.

Well in elder scrolls lore there is many strains of vampire and curse of vampirism. Daggerfall vampires were more like the d&d undead type of vampire. However Morrowind, Oblivion, and Skyrim and online vampires are not. The condition turns the person while they are still living not when they are dead. Normal healing does not work on undead like zombies, however it does work with oblivion, skyrim and online vampires.

Basically Vampires are undying/unliving in elder scrolls lore. In between living and dead, them being healed by energy that heals the living shows that vampires are a form of positive but dark/night aligned deathless if going by d&d standards. Its moral alignment as a vampire might determine how weak they are to things holy just a personal theory.
While an extremely evil aligned vampire might not be able to enter a chapel while a good or neutral aligned one could most likely enter.

The divines likely won't care about the Vampirism so long as your following their mandates or maybe if your a powerful demigod being like the dragonborn which case they still grant their blessing no matter what horrible things you do.

As for alignments and them not being able to be good as some people might think. Well sorry to burst any bubbles but there is several examples of good aligned vampires in the game. House Ravenwatch by its code of Conduct is Lawful Good.
There is a one group of vampires that you need to get the blood to save one of the leaders. The one group obviously good aligned and the other vampire you could get the blood from is evil aligned.

As for Templars if they are following Stendarrs mandates of mercy and justice. They would still keep their Templar abilties if he granted them to the Templar. But then again what they use is light, sun and healing. Maybe not drawn from the divines at all. But given through training they are drawing from the elements and maybe not directly through a divine.