Hmm, I don't know if I'd say I have any favorite parts, and the ones that stick out in my mind are too spoilery to share. But my favorite character by far is Yang Wen-li, honorable mentions going to Oskar von Reuenthal, Julian Minci, and Katerose von Kreutzer. I have favorable opinions of just about every character, and at the very least like all of them for their roll and the dynamic they brought to the series. Also, I'm aware of those OVAs and movies, but I was so burnt out from binging the whole of the series as it was, and with school coming up I decided to defer those for later. If I ever get an LoGH itch, I'll see if any of those OVAs or movies are worth checking out.
There are two things about the anime that I didn't really like. The first is that the directing can be really flat sometimes. Things move at an almost robotic-like speed, which is nice since the plot is constantly moving forward, but at the same time I wish the anime would just slow down and make things more "dramatic" when it's appropriate to. I understand that with a limited musical score it's difficult to draw out certain tense moments, but sometimes it feels like scenes just transition to the next without any sense of artist storytelling. It's probably hard to imagine what I mean, but it's like having someone with a monotone voice read a poem. Obviously it's not good to have someone super over-the-top milk every line like it's the most emotionally riveting thing ever, as that would be unnecessarily cringey, but some artistic liberty in the delivery wouldn't hurt. The second thing I didn't like was that the narrator would strait-up spoil later scenes in the anime. The narrator tells the story from a future perspective, like a historian telling a "legend" from years gone by, which adds to the documentary-like aesthetic, but sometimes he insinuates things too obviously. It's good to weave foreshadowing into the narrative naturally, to give people "aha!" moments when they rewatch it, but to have the narrator say certain things so on-the-nose just detracts kills the mood. This seemed to be a problem more so in the later parts of the anime than the beginning.
Hah, I don't believe I heard any portions from The Nutcracker, otherwise I would have surely remembered. The thing is, I don't actually think I've actually ever watched any anything that handles politics quite like LoGH, though I guess Game of Thrones comes sorta close? LoGH in the broadest sense is an exploration of the conflict between two entities, the authoritarian Galactic Empire and the democratic Free Planets Alliance. That backdrop may seem corny, but the show doesn't make a strawman out of either. Both sides are presented with their benefits and flaws, such as the corruptibility of democracy and the efficiency of central control, which sets it apart from other things I've watched. GoT is similar-ish, but doesn't put politics as a literary theme in the spotlight in the way that LoGH does. If you ever watch LoGH (since I believe you've already seen at least some of GoT), you'll know what I mean when I say they're similar but still markedly different in ways.
I haven't watched Code Geass in years, so I only vaguely remember it. This was back during the days I watched anime almost exclusively for the cute anime girls, so my judgement of the show boils down to "it was fun I guess and Kallen is best girl." I can kinda see the similarities though. From what I remember, Lelouch in Code Geass is trying to change a corrupt authoritarian regime from the inside out, a regime that's at war with a rebel faction. That's similar to Reinhard von Lohengramm's aspirations in relation to the Galactic Empire, and the Free Planets Alliance that they are warring with just so happened to have been formed decades ago as a result of a rebellion. Both are also strongly motivated by their desire to protect/save their sister (younger in Lelouch's case, and older in Reinhard's) for one reason or another. Those are the only spoiler-free comparisons I can draw, but even then the execution and the overarching themes of the anime are drastically different. They just seem to share the skeleton of a premise, and that's ignoring all the Geass power stuff in Code Geass and the whole of the Free Planets Alliance perspective in LoGH.
There were some violent, bloody scenes in the anime, but nothing gory or gratuitous. Nothing too graphic, and seeing as you've found your way on this forum, it's nothing I'd think would bother you. Those scenes were also few and far between, which is why people don't usually mention them when talking about LoGH. As for main character deaths, there are quite a few characters that would be considered important or even main that are offed during the show.
I liked the arcs all pretty equally, and I don't think I have a favorite. All were pretty good in their own right, though the consensus is that the first is probably the weakest since it exists to set the scene, tone, and pacing for the rest of the show. It's not as high stakes as the other arcs, but it's not bad either. There are definitely moments spent for "slice-of-life"-like moments, but I don't know if they're as substantial as you're looking for. Like, there are never entire episodes devoted exclusively to the cast just chilling and being care-free. Every episode is mostly driven by politics and war, with a couple of scenes spent developing characters in more laid-back scenarios sprinkled in there for a brief change of pace.
I feel like the current model of how anime is made makes it difficult to have long-running shows anymore. Aside from the shows that are already established long-runners, most new anime these days rarely get more than two-cour confirmed at a time, with the vast majority falling in the one-cour range. Maybe it was just like this back then too, and LoGH was a freak accident, but then again I'm no anime historian. I'm hoping the best for the show though, since I'd like to see their take on some of the later scenes, even if the character designs are really throwing me off right now. Also, thanks for the clarification on the whole Gundam The Origin and Mobile Suit Gundam situation, that's interesting to know! I'm not at all familiar with the Gundam series, so you'll have to forgive me for my ignorance.