Why? MAJOR SPOILERS WITHIN!

[leader of mortals]

This is how I think it happened...

__ Ellone is born
__ Ellones parents die
__ Ellone is sent to the orphanage
__ Raine adopts her
__ Laguna meets Raine
__ They get it on
__ Laguna leaves for war before Raine realizes she is pregnant
__ Raine has Squall and dies in Labor or shortly afterwards
__ Ellone, already knowing of the orphanage, goes there with her new brother
__ Ellone gets to the age of 18 and leaves to find Laguna. She tells Squall she will be back
__ Squall stays at the orphanage until Headmaster Cid takes the entire group in as SeeD trainees
__ Ellone hears of this and goes to the garden, where she finds Squall at the beginning of the game. Sometime before this, she learns that Laguna gets into trouble with something in the past
__ Ellone uses Squall to save his father and finds it better not to tell Squall of this so his actions in the dream sequence are what he thinks is correct

Seems fine to me...

[Ouch!]

I don't think Squall and Laguna knew that they are related to each other, especially not in the beginning. Maybe they have found out that they are related after the time compression.
If Laguna knew that he was Squall's father, he would of have told him at some point in the game, he didn't even tell him why he ended up being in the orphanage.

It's made clear in the game that Laguna, Kiros, and Ward are all aware of Squall's parentage, specifically if you talk to them on the Ragnarok before beginning to the assault on Lunatic Pandora. Laguna tells Squall that they need to talk when it's all over. Kiros tells Squall it's a good thing he looks like his mother. Ward does his little silent thing, and Kiros translate to something like, "Good thing you don't look like your father." They all know that Laguna is Squall's father. My only guess is that they do not wish to add the emotional burden of telling Squall outright to everything he already has to deal with (i.e. saving the world).

[Takero]

I don't think Laguna Knew at first That Squall was his son . But when Ellone left the white SeeD ship and went to Esthar she told Laguna all about Squall being his son .

[Big D]

This is how I think it happened...

__ Ellone is born
__ Ellones parents die
__ Ellone is sent to the orphanage
__ Raine adopts her
__ Laguna meets Raine
__ They get it on
__ Laguna leaves for war before Raine realizes she is pregnant
__ Raine has Squall and dies in Labor or shortly afterwards
__ Ellone, already knowing of the orphanage, goes there with her new brother
__ Ellone gets to the age of 18 and leaves to find Laguna. She tells Squall she will be back
__ Squall stays at the orphanage until Headmaster Cid takes the entire group in as SeeD trainees
__ Ellone hears of this and goes to the garden, where she finds Squall at the beginning of the game. Sometime before this, she learns that Laguna gets into trouble with something in the past
__ Ellone uses Squall to save his father and finds it better not to tell Squall of this so his actions in the dream sequence are what he thinks is correct

Seems fine to me...

Ellone's parents lived right next door to Raine, so she'd have adopted Ellone without the orphanage being involved. We also know, from the game, that Laguna only left Winhill because Ellone had been abducted and taken to Esthar...
I'm pretty sure it's all detailed in the game, but it's a lot of minutiae really. What matters most is that Ellone would've brought (present-day) Laguna up-to-speed on what Squall is all about.

[Takara]

Ellone and Squall are not related by blood in any way, shape, or form. I think that there may be some confusing occurring from the fact that Ellone was raised by Raine; however, Ellone was only a foster child of Raine because her parents were killed when she was a child for refusing to give their child up to the government.

Laguna arranged for Ellone to be taken back to Raine after the events of the flashback in Esthar; soon after, Raine gave birth to Squall and passed away, and, subsequently, Ellone and the newborn Squall were sent to the orphanage.

Fixed.

Roogle's got it right. As for whether or not Laguna knew Squall was his son, the conversation in the Ragnarok implies he does. I'm not sure if he knew all Squall's life that he had a son or if Ellone told him after she left the White SeeD ship, though. That is never specified in game.

[Innes]

Raine died after Squall was born. Laguna was in Esthar at the time, overthrowing Adel and rescuing Ellone.

There was no parent for Squall, so he was sent to the orphanage.

[champagne supernova]

Here's a quote which should help:
Ellone: "You can't change the past. I just found that out. When I was kidnapped, Uncle Laguna went on a journey to find me... But because he did, Uncle Laguna wasn't able to be by Raine's side when she died. Rained wanted to show Laguna her new born baby... Raine kept calling out for Laguna. So no matter what, I wanted Laguna to stay in the village... But it didn't work."

But, I think Ellone was sent back to Winhill before Raine died. Laguna says: "It was a mistake to send Ellone alone back to Raine. Raine died, and Elle was sent to the orphanage. If I had only gone to Winhill with Elle... I would've bee able to see Raine one last time. Raine was dead. Ellone, missing. My job kept me busy. I was left here thinking about this and that and before I knew it, all this time had passed. Well, that's about it."

Also, the scene where Laguna is at the orphanage is before he rescues Ellone from Esthar. Edea asks Laguna about "this Ellone", to which Laguna replies:
"She was kidnapped by Esthar soldiers. I've been travelling, tryin' all I can to get inside Esthar, but...".

Putting the last two quotes together, it would seem that Laguna only found out he had a son when he saw Ellone again. The timeline should go something like this then:

----Ellone's parents die
----Ellone adopted by Raine
----Laguna and the Lunatic Pandora problem
----Raine rescues Laguna
----Raine & Laguna make Squall (a bit of an euphenism there)
----Ellone kidnapped by Esthar soldiers (this could have
occured before the making of Squall)
----Laguna begins his search for Ellone
----Goes to the orphanage looking for her
----Enters Esthar, saves Ellone, removes Adel
----Sends Ellone home, becomes President
----Raine gives birth to Squall, dies shortly afterwards
----Ellone & Squall sent to the orphanage.

[Cloudane]

Yep that last one is how I recall it.

Plus the following:

-- Laguna didn't seem to give a rats ass that his step-daughter was stuck in some orphanage as he was too busy playing President.
-- Ellone went to find him and tell him everything (including about Squall).

[champagne supernova]

A bit unfair on Laguna. Edea sent Ellone on a boat shortly after the orphanage to protect her. So, no one really knew where she was. He did know she was missing, and he sent ships to look for her (I mean, that's how he found her). He is a bit of a prat nevertheless.