Knowing your horse can die, can you ride it to death or is it just at risk when taking damage?
I'm actually not sure about this. Your horse does have both health and stamina, but I've always kept mine well-maintained enough that neither drops below the minimum. I did lose one horse because I rode him off a cliff, though. You can buy (expensive) "horse revivers" to bring them back to life, but if you don't have one on you or you don't do it fast enough, he/she is gone for good.
The official guide only mentions hunger and dirtiness affecting its health/stamina.
Can you be so honorable and famous that it balances out some of the social negatives of being super dirty, or the other way like you are so well groomed and dressed that many average people still respond positively despite you being a complete psycho?
I believe being well-maintained has a bigger impact on NPC's reaction towards you than being honorable, in terms of pure social interactions. What you're wearing also has an effect on NPC reactions - wearing a bearskin outfit may cause some people to make fun of you, for example.
Is there a level of hygiene that negatively affects your cores/cause illness/etc or is it mostly just social scorn and increased fights?
Nope. You can be poisoned and get sick in the game and that makes you lose your cores faster (as well as being too hot/cold), but hygiene isn't the cause.
However, staying bathed not only affects social stuff, but it also restores your cores.
I've seen several mentions of the lowest honor tier affecting the environment including the weather, but absolutely zero descriptions of anything other than changing some cut scenes, social interactions, bonus outfits/store discounts, and the music, does it affect anything else?
No clue. I've played with exclusively high honor so far. I hate being a jerk to people in video games.
I'm going to play a lengthy session again tonight, and I'll test at least a couple of these things for you, specifically the honor/hygiene impact on social interactions and riding your horse to death.
I will say, I'm not so sure about honor affecting the weather. I'm not doubting it entirely, but the guide doesn't mention anything like that (Granted, it doesn't mention much at
all about how honor affects the world, just what the "game-y" results are).