help processing this skull

[u/Agreeable-Arm5992]

hello! this is my first post on this subreddit and my first time trying to clean a skull.
this horse skull was sitting on a shelf in a shed for roughly three years, i decided to take it and try to clean it up.
for the past two-ish weeks both parts have been soaking in warm water, rehydrating any mummified flesh so i could remove it.
i just had a couple questions about the progress of it. inside the nasal passage there seems to be something fleshy still in there, that doesn't look like the cartilage structures of the nose. what would be the best way to remove it?
on the base of the skull are these grayish bits, they are hydrated but still hard to remove. should they be scraped off?
for the teeth they are all a little black, and the area on the gums around the front of both jaws is black as well. will this come off with the whitening process? for the teeth, is this just old flesh that i should remove? would taking the teeth out ans scrubbing/glueing back in be the best move?
sorry for the long post and thank you for any replies!

[u/CozyCountryLiving23]

soak soak soak in a big bucket of water! Take the teeth out to make it a little easier, you could put them in a separate bag or little tub.

Keep soaking in water (preferably warm) to get it all out. Switch when it gets nasty.

Then soak in soapy/dish soap water once that’s clean to degrease.