Yes, more or less! I agree with most of your conclusions. Keep in mind though that the earlier seasons only exist in the form of outlines, and it should be possible to include info and reminders about this stuff when/if they're turned into scripts.
Thinking about this, having had the explicit prophecy a few seasons ago wouldn't do us much good anyway...we'd have to have a "Remember this? It's finally relevant now!" moment because it was so long ago, and considering how many times Huan has fought without worry already bringing it up now would be like showing Chekhov's figurative gun and doing nothing with it for a long time, and then before it's finally used someone reminds everyone that the gun has been there the whole time.
It would be better to just have a moment clearly establishing that in regards to the Doom of Mandos, Huan is firstly a character affected by it - not just a pet who Mandos isn't including in the group of people he's adressing by default - and secondly has agency and moral responsibility, who makes a conscious choice to stay faithful to Celegorm and knows he's dooming himself in the process.
I'd portray this first in Episode 7 of Season 3 by showing him listening (as much as you can make that clear with a dog) to the Doom, and then lingering and looking back for a bit before he turns around to catch up to Celegorm and the others on his own accord (without being called or looked for).
In the script outline for Episode 11 of Season 3, when the Valar discuss whether they should intervene in Middle-Earth, Yavanna doesn't want to abandon Middle-Earth to Morgoth. I'd add to that by having Orome championing one of the opposing arguments (the "It's their free will to fight Morgoth" one would probably fit best), to which Yavanna replies that Orome's chief hound has fallen under the Doom of Woe out of loyalty and is in danger if they don't help the inhabitants of Middle-Earth. Orome replies that Huan chose to stay loyal to Celegorm on his own (which is important considering he will choose to abandon Celegorm for Luthien later on), and that his willingness to pay this price secures him a part in the songs (the ones Feanor mentioned in his reply to Manwe and were brought up at the beginning of this scene).
I think we might want to consider introducing Carcaroth next season. It was discussed briefly here:
DISCUSSION ABOUT CARCAROTH'S INTRO (posts #208-#239 I believe)
Thanks for the link!

I don't really mind introducing him in Season 6, in fact it's probably best to keep him secret until after Sauron loses to Huan; that Huan manages to defeat Sauron and Draugluin is a great eucatastrophic moment in the story, which would be robbed of surprise and tension if the viewers know that Huan can't die here and Sauron has to fail because he lacks crucial information.
The important parts that we need to set up for the villains are that (1) Morgoth knows about the prophecy when he starts to "prepare" Carcharoth in whatever way we see fit, (2) Sauron knows about the prophecy when he turns himself into a wolf in Tol-in-Gaurhoth but (3) doesn't know that Morgoth is preparing Carcharoth, because it makes assuming wolf-form himself pointless.
Personally, I'd like to do (1) and (2) before the Dagor Bragollach because Huan becomes pretty irrelevant to the villains after that until he shows up at Sauron's gates. And starting at that point means having Sauron figuring out the prophecy as Draugluin is dying, getting that information to Morgoth somehow (I wouldn't know how without introducing a new convienient witness at Tol-in-Gaurhoth since everyone there dies or flees somewhere else), and him getting Carcharoth ready before Beren and Luthien show up at Angband. That is firstly too much and too convient to happen so quickly, and secondly showing Morgoth (and Carcharoth) a lot right before Beren and Luthien meet him would take away from his role as the distant, ultimate menaces in Beren's quest and the moment when they finally meet him in the throne room.
My current intuition on how we should do this plotline is that at any time before the Dagor Bragollach, Thuringwethil spies on the fortifications at the Pass of Aglon to kill two birds with one stone - figure out the fortification's weaknesses (so they can get behind the main Feanorian force at Himring and Maglor's Gap; also, the Pass of Aglon is the weakest point apart from Maglor's Gap which doesn't have many secrets to discover - it's just flat land with lots of troops) and Huan's weakness (because he is part of the Doomed that came from Aman, but has been a mighty wolf-counterforce in Celegorm's cavalry, apparently without ever getting in serious danger). Celegorm and Huan have an insight about Huan's doom to die before the mightiest wolf (if we want to, with a connection to Fingolfin's assault proposal and/or Aredhel's dissapearance/death, since she is Celegorm's friend) in more or less vague terms. He talks about it with words, again in more or less vague terms depending on our preference, with Huan and/or a companion of his (probably Curufin), and Thuringwethil overhears that. After her task is sufficiently complete, she goes to Sauron and Morgoth to relay the information she gathered (the less clear the prophecy was to her, the more the villains need some insight of their own, which would be fine with me). Morgoth either doesn't react to it or only in the vaguest of ways, while Sauron prepares Draugluin to seek out Huan in future battles - that would be a good plot element to build on during the Dagor Bragollach and Huan's retreat along with Celegorm.