First of all, I am out of likes. But I like all this. I feel the need to thank you all. I very much enjoy this sort of reflected discussion. (Just... don't tell a few of the others I said that, I have a reputation to lose here.)
Were the Wardens in DAI hearing the Calling because the OGS was trying to rally the Wardens away from Corypheus and his demons?
Edit: As for where the OGS from the previously slain Archdemons are.... it certainly would be fun if there were another 4 gods walking around Thedas right now!
I personally just cannot imagine the soul would die like that. But more than this I don't see how else they want to figure Kieran in. They cannot just make two games, one with and one without Kieran. But they cannot simply ignore Kieran either. (Well, they could, but that would not live up to BioWare's reputation for delivering good writing.) So the soul he harbored must become influential at some point. And they need a way to make the same thing happen for players who have no Kieran. I consider letting the OGS survive the slaying the "simplest" approach.
Hmmm... if archdemons aren't in fact Old Gods and are just their bodily prisons, then how were the Old Gods sealed into them?
I am fairly sure that is exactly what Morrigan tells us, that the archdemon is not the OG, but only has an OGS trapped inside. If madrar were here, she could explain. She knows all the facts related to this. This way, both sides are willing: the OG wants to be deliberated, find some host to leave and get back to the world. The Darkspawn (in that case a dragon) is a mindless beast controlled by the song and is willing to become that vessel. Hence, following the idea Sister Squish pointed out before, we have the same thing as with Flemeth. A willing soul is transferred to a willing body.
If the OGs are not tainted, they cannot access the Taint Hive Mind and order Darkspawn around. Which'd mean that the song Darkspawn hear while searching for the OGs is different from the song produced by the Taint. But why would they hear it, then? Because of the Taint? Maybe the Taint makes them misinterpret an untainted OGs song and taint him instead of staying away?
Why do the darkspawn hear the Old God's Song in the first place? The Old Gods are supposedly not tainted until the darkspawn interact with them while opening the cage, possibly by using blood to open the prison.
I could come up with a theory from cognitive science here but I will refrain from doing so because last time I tried flabbadance totally did not understand what I tried to say. 
So I will only point out that we know nothing of "the song" or even "the taint" or "the blight". Maybe there is one universal song, a sort of wave that touches all things. But some are blocked from noticing it (strong mind and such) while others are vulnerable (tainted creatures). So they all react. If we go a step further and assume that the OG themselves emit the song, we have a sound source and the receivers (those with tainted blood, maybe also other gods etc.) approach it.
Let me compare it to our previous topic of Nazi Germany, maybe. Imagine Hitler had not been a racist himself, so what? All he needed to do is feed into the common racist mood (hive mind) at the time and use it to further his goals. I think the same could be true for the OGS. They might not be tainted themselves, but they emit a song that can influence (or create) the hive mind in order to gather their "army".
I think it is very hard to tell any relations and influences while not knowing what exactly the song or the taint are in the first place.
And what pattern is that? 
You like the strong outlaw male, the underdog (good or evil does not matter because it's all only different shades of gray anyway) who fights against all the odds for whatever he believes in, preferably doing so with eloquent and sophisticated means, possibly commanding or manipulating others, and always with a big plan in mind. A plan that reaches beyond "We must slay archdemon so we be happy again".