I found a fanfiction that fits some of the details mentioned here, but doesn't fit many of the others. I'm posting it as an answer in case you may have misremembered some details, or combined two (or more) stories. I will address your bullets one by one, noting where the details match and where they don't match. The story is The Dark Phoenix by Lordstar016.
Harry meets Rowena Ravenclaw's familiar, which was either a female
eagle or a raven. The familiar is the guardian of Rowena's hidden
library and decided Harry's level of access to texts. The bird tells
him that Godric's familiar (a lion that I am almost positive Harry
can wear as a tattoo or item on his body) and Helga's (a huge female
badger which I'm positive ended up being in the Room of Requirements
the whole time living in a tree and revealed herself to help cure an
illness) are aware of him and will make themselves known in time.
There was some element of him being "chosen" by them.
Harry does meet Ravenclaw's familiar, it is part raven and part eagle, and can loosely be considered the guardian of Ravenclaw's library:
The trio entered the right hall and entered a HUGE library. Books were everywhere and many many scrolls and other items that held knowledge. It was bigger than the main library in the school.
Suddenly a large black bird flew down towards Harry. Harry got a good look at the bird.
It's wingspan was about the size of Gondor's body and its body was pretty thing. It was black with silver tints on its feathers. It was a female and had grey eyes and a sharp beak. It looked like a overgrown Raven but it had the talons of a eagle and the speed of a hawk. It looked majestic as hell and it landed right in front of him. He was only 1 inch smaller than it. The creature Harry knew was called a Birvan a large mixture of several speices of birds.
"You must be Rowena Ravenclaw's familiar am I right?" asked Harry.
The bird nodded.
"Yes your correct sire I am Ravana." the bird said.
"Of course this is The Study of Wisdom Rowena Ravenclaw's secret chamber." Ravana said.
However, there is no mention of the bird deciding Harry's level of access to texts. In fact, in the rest of the story Harry never even uses this library.
Gryffindor's familiar is also in the story, and while it is a griffin rather than a lion, it is described as a lion (my emphasis):
Harry turned his head and by the main entrance to the room they were in was a giant beastly looking lion thing. It was a male and had long sharp teeth about the size of small swords and giant muscular legs and paws with long claws on the end of them. Its tail was thin but flexible looking and its mane was dark red. Its eyes were sky blue and its fur was golden. The only thing that didn't make it a over sized lion was the fact that it had some feathers on its back and legs a bit and had giant eagle wings. It was about 30 feet tall. Harry was up to its chin however its width was larger than him big time and it looked like it could swallow half of Harry in one gulp. Harry knew what this creature was. It was a Griffon and it looked like it was ready to kill.
"How... how can I understand you?" Harry asked.
"Because Young Harry he is Godric Gryffindor's familiar as I am Salazar Slytherin's familiar." Sarota hissed out.
Similarly, Hufflepuff's familiar is a "huge female badger":
It looked like a oversized honey badger but yellow and black fur and large long body. It was a female and as shorter than Harry when walking on all fours but was 1 inch taller when standing on two legs.
"Hello you must be Helga Hufflepuff's familiar whats your name?" asked Harry.
"My name is Halen dear." she said in a kind tone.
However, it was not Ravenclaw's familiar that told Harry about Hufflepuff's and Gryffindor's familiars; rather it was Slytherin's familiar that showed Harry all the other ones. Additionally, there is no mention of Hufflepuff's familiar living in a tree in the Room of Requirement; instead it lives in Hufflepuff's secret chamber. Nor did Hufflepuff's familiar have anything to do with curing an illness. Also, Harry was not "chosen" by them; rather, he was the heir (of all of them):
After a few minutes as piece of paper shot out. Sharptooth read it out loud.
"Harry James Potter"
Son of James Charles Potter = Deceased and Lily Potter nee Evans = Deceased
Born: July 31st 1980.
Grandson of Charles Alexander Potter = Deceased and Dora Alina Potter nee Black
Heir: Potter = Main Line = direct
Black = Chosen Heir = chosen
Peverell = Father's side = direct
Gryffindor = Father's side = direct
Slytherin = Mother's side = direct
Ravenclaw = Mother's side = direct
Hufflepuff = Father's side = distant
Gaunt = Conquest = conquest
Harry had already killed Slytherins familiar in the Chamber based on
a conversation I remember him having with Ravenclaw’s familiar, so
this story began post Chamber of Secrets (this may be wrong as
slytherin a familiar might have been a different basilisk not the one
in the chamber).
Harry hadn't actually killed Slytherin's familiar, but he (and everyone else) thought that he had. Slytherin's familiar was the basilisk in the Chamber of Secrets (named Sarota)but it turned out that Harry had wounded it but it had survived:
Harry turned to him. "Sarota can enter the chamber and leave as he pleases he was after all Salazar's familiar."
The hall was in an uproar. "I thought you said he killed it!" screamed many students. While certain Slytherins and Ravenclaws remembered the name from before. "EVERY ONE PLEASE CALM DOWN!" yelled Harry. Everyone stopped to look at him. "Alright yes the Headmaster said I killed him and during 2d year I thought I did as I had stabbed him threw the head and I thought I hit his brain. However I merely knocked him unconscious and made him wounded. I went back in the middle of third year and found him and we made peace... kinda." Harry explained. The hall still was frightened about there being a 60 ft basilisk in the school but they continued to listen to Dumbledore.
In any case the story does begin "post Chamber of Secrets".
Very loosely I recall that Harry may have stumbled onto some hidden
space of Ravenclaw’s in the Castle which led him into initial contact
with her familiar.
As mentioned in response to the first bullet, Harry found Ravenclaw's familiar in a secret chamber that Ravenclaw had in the castle.
Dumbledore has been siphoning power from Hogwarts's wards. Harry
follows him to a tower and confronts him during this act. I Roughly
remember Dumbledore is removed from Hogwarts and returns with
Ministry Aurors, etc.
There is no mention of Dumbledore siphoning power from Hogwarts's ward, or any confrontation relating to this on a tower. However, in the story Dumbledore is not quite good, as it turns out he had been obstructing Harry's parents' will, enabling theft from Harry's vaults, placing magic blockers on Harry, and creating an illegal marriage contract between Harry and Ginny:
"Molly, Ginevra and Ronald have been taking money from the Vault since 1980 up until this year. Also Molly has made a new account under the name of "Prewett" for herself which in turn means she steals even more money. Albus Dumbledore was the one who gave them the key to Mr. Potter's vault due to him being Mr. Potter's magical guardian and after reading the will of his parents had NO right to claim. Also Albus Dumbledore put so many blocks on Harry's magic that it was nearly all blocked. Then finally to top it all off he made a illegal marriage contract with Ginevra Weasley and Harry Potter." Sharptooth said in one go.
Because of these misdeeds, there was a discussion about removing him from his position, but that did not occur by the end of the story:
"As for you Albus you better hope you got a pretty good lawyer for I think you might end up getting a lot of your positions taken away from you." said Sirius glaring at the man.
There is, however, mention of a confrontation on a tower involving unnamed people, one of whom may have been Dumbledore:
Harry saw something in his mind. It was a vision. It showed a army charging at another army. A cloaked man battling two people on a tall tower.
However, this is never elaborated upon later in the story.
Dumbledore practically destroyed the original wards around Hogwarts
by using the instruments from his office to carve slices out of them
essentially, but Slytherin had designed a secondary ward scheme which
Harry activates. This makes me think there was a heartstone or Ward
stone scene but it’s to fuzzy.
There is nothing remotely resembling this in the story.
There is a tower that is for Harry as an heir to the founders or
founders chosen whatever he's called. Dumbledore managed to find it
at one point and shattered the guardian status of the founders
familiar as I recall there’s a scene where Harry rebuilds this.
As mentioned in response to the first bullet, Harry was indeed the heir to all the founders. But none of the other details of this bullet are present in the story. There is a network of secret chambers that Harry has access to as their heir, but it's not a tower and Dumbledore never found it or did anything to it.
One particular scene involved Rowena's familiar discovering the
diadem was a Horcrux and going absolutely batshit. I remember they
used Magik with the K spelling fairly often.
There is no mention of the diadem or Horcruxes, and I did not notice any instances of "Magik" with a K.
I vaguely recall the familiars had some sort of shape shifting
ability, or were something other than the animals whose forms they
assumed as Rowena and Godric's familiars had a romantic past of sorts
they were discussing telling Harry about in one scene. Also
Ravenclaws familiar become the Transfiguration teacher after
McGonagall become Headmistress and she had a human female form.
There is nothing of this in the story.
There is a close relationship between Harry and Severus but it's not
slash and I can't remember if it's father/son, or simply
professor/student. Based on another comment Harry and Draco begin
some sort of relationship but I can’t recall the details ie Veela
There is not much of a relationship described between Harry and Snape. There is one point where it might be implied that their relationship will become better, but that is never shown to actually happen:
Severus Snape however was silent and white as a ghost. James Potter his life long enemy had tried to apologize even going as far as to begging for forgiveness. Severus felt like a child. He had let a grudge rule over his emotions about a boy. Lily's boy. He felt sick. So sick that he looked paler than usual. He felt his whole world crash down and he couldn't stop it. He let a single tear fall out of his eye.
There is also no particular relationship shown between Harry and Malfoy.
I know it's not a lot to go on but this has been tormenting me and
I’ve wracked through about a hundred pages on FanFiction.net trying
to find it. The title was something completely ubiquitous as I
remember not expecting much based on it and being completely
engrossed and thinking it should have a better name. It was a
relatively lengthy fanfic, so at least 100k words but possibly much
more and, it was incomplete on my last read. I am 97% certain this
was on FF.net but I have read the odd story on Archive of Our Own, so
it could be there.
The story is indeed on fanfiction.net. The title The Dark Phoenix is indeed pretty vague and does not really describe the story as it currently stands. It is ten chapters, if that's considered relatively lengthy, and indeed it appears to be incomplete.
All in all there are quite a few details that do not fit, but perhaps there is still a chance that this is the story you are looking for.