For the purposes of this analysis (to keep things simple) I will refer to the singer as W (woman) and the singer's lover (who I believe is the intended audience) as M (man). I'm going to do this in a few installments, because otherwise it's ridiculously long.
Maybe there’s a God above
But all I’ve ever learned from love
Was how to shoot somebody who outdrew you
It’s not a cry you can hear at night
It’s not somebody who's seen the light
It’s a cold and it’s a broken Hallelujah
More questioning of faith. W (the singer) is apparently not enamored with the religion set out in popular doctrine. This is also addressed in the next line, when she references Church teachings that "God is love" and such, saying that she didn't really learn anything useful from these teachings except how to duck out of a relationship before getting stung. A "quit before they can fire you" mentality. Somewhat defeatist, in my opinion. Then again, the whole reference to love and learning could have to do with the more immediate experience of romantic love, and implies again that she's been hurt in relationships before and is not eager to undergo more pain. Again the line "a cold and... broken Hallelujah," presumably used to describe "love," whatever that is. The lines immediately preceding are just further emphasizing the last line, crying that love is not benevolent and transcendental. (I'm feeling wordy today; can you tell?)
You say I took the name in vain
I don't even know the name
But if I did, well really, what's it to you?
There's a blaze of light
In every word
It doesn't matter which you heard
The holy or the broken Hallelujah
Again this seems a religious reference. Perhaps to the hypocrisy of religion, "teaching" love without feeling it. This verse really resonates with me, as it describes the individuality of faith. W's concept of divinity is different from M's, so her "name" is not his. Furthermore (in the third line) why does he care? Her faith is her own, her ideas personal, and her thoughts her right. The next two lines suggest the power of language and its ability to express feelings and ideas beyond just consonants and vowels. The last two lines seem like a dismissal of M; "it doesn't matter to me what you think because you're outta here."
I did my best, it wasn't much
I couldn't feel, so I tried to touch
I've told the truth, I didn't come to fool you
And even though
It all went wrong
I'll stand before the Lord of Song
With nothing on my tongue but Hallelujah
W is tired of arguing, and it really doesn't matter anyway what M says. She's done. She'd fallen out of love before this, but for the sake of harmony she held onto the relationship. Because of that she has no regrets, and she can stand before the Lord of Song without shame. The "Lord of Song" could refer to a god, to Death, or just to some amorphous ruling power (like human conscience), so it's a reference to a sort of judgement by that higher power. And she will stand dignified.
Is W's whole hang-up on religion just a metaphor for her romantic relationship with M? I think it could be interpreted as such, and W's rocky relationship with divinity as a metaphor for her crumbling relationship with M. Or the other way around, I suppose. Either way, I still very much like the song. Thanks for reading!