Addressing these point-by-point...
Any sense that Micheal Burnham being raised by Vulcans meant anything regarding her personality -- Maybe.
Any idea that Burnham had Klingon-related PTSD and/or might be a bit racist towards Klingons -- I never got the sense she has PTSD. Just remorse.
The tight narrative focus on Burnham. She used to be The Very Special Protagonist who solved everything and saved the day. Now she just leads the "A plot" each week, and is being led around by the nose by Georgiou and Lorca ATM -- At the moment.
Saru having any character traits whatsoever. He used to be a coward who was resentful towards Burnham, which was shitty characterization, but it was something. Now he's just a generic XO with no personality - only saved by Doug Jones's performance. -- He's growing into the role of having more responsibility. He's the XO instead just someone who wants to be. He does have a personality. He's a bit too stiff and proper. He's also had to cope with making harder and harder choices. Major progress that he no longer feels the need to compare himself to other Captains and has finally begun to accept Burnham.
Tilly changing from a very weird, possibly ASD character to just slightly adorkable -- She's come out of her shell maybe for the first time in her life. She probably didn't get to socialize much before and probably felt like an outsider at the Academy and probably when she was growing up. I know because I used to be like this, until college.
The mysterious black badges and floating water seen in the third episode -- Yes. That's disappeared.
The idea that Discovery is a special science ship researching all kinds of black-ops tech, rather than just the spore-drive machine -- The jury's still out on that one.
Stamets being a grumpy jerk -- He didn't like being assigned to Discovery at first. But then he found his niche. And he found Culber. That made him happy. Those make him not be a grumpy jerk anymore.
Klingons speaking in their own language with subtitles - The last time we see a Klingon ship was in "Into the Forest I Go" where it makes sense that Klingons would be speaking in subtitles because they'd be surrounded by other Klingons. We haven't had that since. So, we have the Universal Translator.
The idea that the 24 houses of the Klingon Empire were going to actually mean something -- It still might. Don't jump to conclusions. Just because Kol's ship was destroyed doesn't mean the 24 houses won't unite. Kol wasn't as important as he fancied himself to be. Plus they went out of their way to mention in "The Wolf Inside" that the Klingons uniting in the Mirror Universe might be the key to them making peace if the Klingons in the Prime Universe unite.
The "lower decks" concept. Originally Tilly was just in the show because she happened to be Burnham's roommate. Now she's doing officer-level work as a cadet just because she's a main cast member and must be doing something. Some of this can be attributed in-universe to character growth, but IMHO that's headcannon, they were trying to tweak the show as each script was finished. -- As she shows she can do more, she's being given more responsibility. That's not head-canon. That's common sense. When she graduates she'll have a leg up on other graduates who weren't doing all this. To date not much has been done with most of the bridge crew. Sure we
see them, seeing them at all is about all we do. I have no idea who the bridge crew is besides what they look like. So it's still a lower decks and non-bridge crew focus.
