Most of the answers are in HF, the novel mostly the movies are not going to go into depth either, but also
I can't find shit without spoilers
Well yes because is supposed to be explained later, stuff about the third magic is interwined with the story, what the root is is not really a spoiler
The things you already should know by this point
Both having the power to grant most any wish, but at an equivalent price. I related it to the behelit from berserk.
Kinda, the only equivalent exchange is how much energy it costs, the rest is just because the grail is corrupted
Gil is incarnated because he was bathed with the mud it gave him a real living body (read not spirit thing like servants that are not really considered alive) the mud doesn't come from the root but from the grail, is the curses
CS are to control servants, you don't need them to form a pact
Medea didn't had infinite CS, she just stole the ones from Shirou
Gil is a servant as far as the definition goes, he just has a body, Kirei does have CS and a lot they show them in Zero and UBW but again it would nlt matter if he didn't
The church decides the supervisor Kirei is the church guy in town he is tasked with that, he got the spare CS and his father was the supervisor that is enough
Also wtf was the church trying to do in zero?
Is pretty clear, they outright said their goal, guarantee Tokiomi gets the grail because thry know what he wants it for and they have a good relation, just that
Also I really hate how they don't explain very well in this series and then have so many exceptions to rules that you have to try and figure out for yourself
The novels explain it a lot sometimes too much, not like there's no things you have to just figure but the info provided is enough
Then the grail apparently has a mind of its own or some shit. And I guess it wants to have another grail war 10 years later instead of 60 because idfk lol
It kinda does have a mind of its own, you already saw it, but it started 50 yeears early because of how the proccess was interrupted, there was still a lot of energy left, it just takes 60 years because is the time it takes to gather mana, if you inject mire it takes less to get to 100%
For the rest just look here
tmdict.com