To clarify, is your basic premise that, once a Cleric has successfully cast a spell using the Initiate of Mystra feat, that spell will carry over into any antimagic or dead magic field the Cleric subsequently enters?
Yes.
Hark back to sentence structure, or even precedence in D&D as seen through my favorite example of Powerful Build. While the introduction says cast which can appear limiting the rest of the entry goes on to describe how this works, what it does, and it ends with "
If this check is successful, your spell functions normally."
And there is no reason to think it keys to one specific AMF or dead magic zone. Actually, could you imagine thinking it would on a grand scale? Instead of dealing with
typeofs you're working with specific
instances. Like imagine Contingency against an attack but the debate is centered around how McFighter Dirtnapper's 3rd attack on the day of May 5th 2020 is not the same as McFighter Dirtnapper's 20th attack on the day of August 9th 2019. And while a statement of that would be technically true, no one - not even the rules - is really concerned or cares.
As for knowing if you succeeded on your CL check. Well it appears you'd find out when you find an AMF. I don't think the caster inherently knows the success/failure of his CL checks like Saves, just his measure of willpower through the metaknowledge of the Player rolling the check or other force-of-effort themed feats (like reserves of strength). You just have to hope your magical might is stronger than the other guy's, see also Kell's post and how he talks about a high CL.