-That red hair chick seems so out of place. She's too... how should I say it, sorority chick out of place? Like her mannerisms don't fit when compared to all the other previous female Starfleet personnel that I can recall. They're all more serious type or other.
-How'd they get the jump drive working again? The tardigrade was released and the only method was injecting the serum into a human which I don't think the engineer guy did again after the first attempt else I would've saw holes in him.
-The captain is too trustworthy of a guy they just rescued to put him in such a high position and not watched for a bit. Even the admiral mentioned that.
Tilly is a young (and apparently younger than most), eager, easily-excitable cadet. I don't recall a lot of young women cadets in the past, but I don't see anything especially "sorority-like" about her, and it's not exactly unusual for past Star Trek cadets to be overeager. Most of them just haven't been female.
Good question about the drive. I suspect they've normalized Lt. Stamets doing it (at least on the Discovery proper), though it would have been nice if they specifically said so during the ep.
Yes, Lorca's been shady from the start and eps 5-6 started to give us some answers as to why that is. It seems his heart is in the right place, but his ethics are clearly not in line with those of Star Fleet or the Federation. He was overdue to be reined in for faking his way through his psychological screenings, but the Admiral that was going to make sure that happens has now been taken prisoner, whoops.
I just found out about the spore drive patch his doc partner gave him on his arm to help with the jump integrations in Episode 7.
I also have issue with them using high ranking officers for front line duties. First one died in the Klingon battle and now a 2nd admiral is captured by Klingons. They sure got enough admirals to throw around or have no adequate protection for them (just shuttle them around or letting them know which ship they're on...). Irks me to the core. Yes I know it was suppose to be the high ranking vulcan Sarek (Spock's father) to be there but he was injured.
Eventually Lorca should get in trouble for using Stamets to run the drive, among other things.
Yes, sending in the commanding officers for just about anything is something I've been questioning at least since TNG aired, but I believe they've been doing it since TOS and I can't imagine they're suddenly going to stop now. (I suspect in most cases it's a convention for giving all top-billed actors maximum screen time.)
In the case of negotiations, I don't understand what you mean. Of course it's going to be someone with a high rank. It's an insult to the other party if you send someone with no real decision making power.
Can't recall when that female admiral went there but the person they met wasn't the commander who later appeared in some hologram after they've been captured. Surprised she wasn't either briefed or know something was amiss when she met the person in question face to face.