"The hollowed out appearance of The Nameless King and its connotations is an element of great debate. He does not speak, a trait shared by Hollows who have lost their sanity, but this on its own is not enough evidence. If he is the Firstborn son of Gwyn, this may be a result of him being stripped of his inheiritance, as though the fire of his lordship were ripped from him, but this is contradicted by his ability to continue to use lightning. It is possible that he was branded with the sign of the undead, as this appears to be the case for all of the warriors that follow him."
Basically? No, Nameless King isn't "clearly" hollow. There are mitigating factors to contradict this- for example, being a god rather than a human. So no.
Your points about the Flame are invalid because of what I did counter- even assuming you somehow proved your point via conjecture, the Flame isn't a soul. If you're just implying "well the flame does x and y thing, ashen carries it, ergo x and y goes on his profile", then... no? Those things ostensibly stop when the flame goes out, or, more accurately, if they continued going is up in the air. So the flame does not provably still do those things whilst in Ashen's possession and, again, it would mean very little in terms of Ashen's own power set, since all it does is constitute time going forward, it wouldn't give Ashen any nifty little timehax.