1. how far removed do you like to take the family tree when it comes to cousins and other relations? Like is there a point where it doesn’t matter like third and fourth cousins? (I say this since you mentioned Dove and Tiger would share a same great grandma/great great grandma, making them second/third cousins respectively which on it’s own isn’t really that bad genetic wise but I’m not sure how you categorize it in the rewrite)
2. If Tigerheart isn’t Tawny’s son, why did Rowan name him Tiger-?
3. How far back do the cat track their family lines? Do they call themselves by a specific family name?
Oh hi, don't worry i type alot too, sorry for the long response, but answering your questions:
1 - Well, i prefer to totally remove that they are related by blood if they are a couple, for more genetic diversity, and for new blood in the clan(as inviting loners and kittypets into the clan), one example i can think of is Heathertail and Breezecloud, since i totally rewrote that Onewhisker and Ashfoot are siblings, so now only Ashfoot and Morningflower are siblings from another pairing, while Onewhisker is the solo kit of Wrenflight and Stagleap.
But Breezecloud and Heathertail would be still tecnically related, IF Heathertail was Ruby's kit together with Darktail, but since she is not, they are not related.
However, i'm still thinking about the Tawnyclaw being the mother of Tigerheart thing, since i don't know if i should keep it or not, but i like the idea of Rowanclaw's kits being not related to Tigerclaw, so we have more diversity in Shadowclan, but i think i wouldn't mind making Tawny their mother in all honesty, as you said, it's not that horrible compared to other canon pairings(*cof* *cof* Ivypool and Fernsong *cof* *cof*);
2 - Honestly, i didn't thought of that when i made the Fireheart's family tree, so maybe i will make Rowanclaw rename him, since i was thinking of adding that Tigerclaw could manipulate Shadowclan before trying to murder Star Bluefur, as a fail safe if he couldn't murder her and that Shadowclan was weak at that time, so it was easier, so maybe Rowanpaw was one of the cats that were mostly manipulated by Tigerclaw, and created a admiration for him, but after Shadowclan discovers that they were tricked, so Tigerclaw can have his revenge on Thunderclan, maybe Rowanclaw wanted to name his kit after Tigerclaw as a "Firestar named Brambleclaw that way, so the clan could move on from the past" way, which i really don't like it.
But i'm thinking that i will rename Tigerheart as Talonheart, for Rowanclaw's apprentice that was hated crimed by kittypets(yes, that happened in canon), since i think he would really want to honour his apprentice, i don't know, i want some opinions here.
I hope i answered you questions^^
3- I think i already said in one of my posts, but i track 'till Mapleshade's Vengeance if possible, since if i'm not wrong, this is the most far back the clans was ever written that was still fairly close to the modern clans, and another reason is the impact that this era made on Thunderclan, since i think i didn't mention it yet(which is a crime), but Thunderclan after the Mapleshade's incident revoked all queen's rights, and the queen's rights would only return in Star Bluefur's reign, this brought so many problems in Thunderclan, since she-cats and toms that have kits outside the clan, would HAVE to name the other parent before the kit is born, so i think it made sense in the rewrite sense, as this era was very important to the clans, so why not start by it? Note: Only Thunderclan made this, the other three still had and have queen's right.
I forgot to answer the family name when i answered this, but adding it later better than never: No, while yes that some cat families have a famous, relevant or antagonistic against the clans's cat, and cats in universe can recall it, with some cats having a prejudice against them because of a family member(Brambleflower), or have some expectation on them from being related to some important clan figure(Silverstream or Whitestorm), but the family itself doesn't really have a name, it's more a way that i can organize myself.
so i thought of this au idea last night as i was falling asleep and spammed da crew about it
TL;DR: Crowfeather gives breezekit to leaf and squirrel after nightcloud dies giving birth. Breeze is raised in ThunderClan, believing the three to be his full siblings. Hijinks ensue.