I know it's because Liquid has practically the same genes as Snake, but if Snake is injected with it, what happens it jumps to Liquid? I don't get it.

How does FoxDie work? Is it not injected or is it just thrown in the air and it chases the one with the genes that the virus is set out to kill?

Help!

Snake was the carrier and it killed more than just Liquid. It also killed Decoy Octopus (posing as the DARPA Cheif, Donald Anderson) and President of ARMS-TECH, Kenneth Baker. We assume that Naomi altered it that way because at the time she was wanting revenge on Snake herself, and didn't want the US Government doing her dirty work for her, though she eventually forgave Snake.

So why does it kill everyone around Snake, but not Snake himself? Like what's the reason for it killing Darpa Chief, etc.

It kills everyone it is programmed to kill. This included Foxhound members and Baker.

As it seems, Naomi also made the virus to kill Snake at a random time, but this didn't happen in the end as it seems like Snake was cured of Foxdie after the Shadow Moses.

You all Chill I'll tell you all about it :D
Foxdie kills who it is programmed to kill. It causes to make the chosen people to create a heart attack even though there is nothing wrong with you. Then they die. The Foxdie that Naomi injected in Snake was a special one. The Snake Foxdie is programmed to strike at any time. There is no time limit or shit, it just kills when its activated. And no one knows when it will be activated. My guess is that the patriots know when it will be activated and when it strikes Snake.

So there you have it, Snakes Foxdie can strike at any time. No one knows.

Oh, so it's not completely gone from Snake? So Naomi programmed it to kill Liquid too? I still don't get how it came to Liquid, and what is the virus, how does it travel, etc? Who programmed it to kill Darpa chief, baker and liquid?

No its not completely gone from Snake. At least what we know of.
Yes it was programmed to kill Liquid too.
Naomi programmed it in order to kill the DARPA chief and Kenneth Baker.

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FOXDIE is a engineered Virus created to take out specific people by identifying the person's DNA and their Nanomachines. It had been in development for years before the Shadow Moses incident, but research hit a wall until Naomi Hunter joined the project. Since then, the virus has been altered and re-created several times, to do several different specific things.

The virus was injected into an unaware Solid Snake shortly before he departed for Shadow Moses. The virus was programmed to kill off the President of ARMSTech: Kenneth Baker, the renegade members of FOXHOUND and the entire of the GENOME Army. Interestingly, DARPA Cheif Donald Anderson was not a part of FOXDIE's programming due to the fact that Jim Houseman was good friends with him and wished to spare his life. Snake also eventually discovered that Naomi made an unknown edit to the virus so that at some random point it would also kill Snake, because he killed her closest relative, Frank Jaeger. Although the virus was successful in killing Liquid Snake, who was genetically identical to Solid Snake, Liquid's death served as indication that Solid Snake would eventually succumb to the virus. Dr. Naomi Hunter, when asked when Snake would die, said "Live, Snake. That's all I can say to you." Since Naomi had set the virus to activate at a "wildcard" value, it is unclear whether or not Dr. Hunter even knows when Snake will die.

In 2009, the FOXDIE virus is re-created into a computer virus by Emma Emmerich in order to disrupt the AI of Arsenal Gear (Codenamed: GW) and subtract the parameters which The Patriots had setup in order to delete any data about them so that Solid Snake and Otacon would discover who they are. The computer virus was part of the S3 program test that manipulated the events of the Big Shell incident into an orchestrated recreation of the Shadow Moses incident.

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Thats all about FoxDie, at least what we all know of.
Source: FoxDie - The Metal Gear Wiki
I know the MG Wiki is not a reliable source but at the current time all the facts make up.

Yeah I see now, so they injected it into Snake in order to kill everyone he comes into contact with, but it wasn't intended to kill Snake until Naomi edited it to randomly kill Snake. I see. And the Colonel didn't know about FoxDie? He just thought Snake would go in there to dismantle the operation etc but not through the means of FoxDie right.

No, it WAS meant to kill Snake, but Naomi edited not to, because she wanted to have her on revenge.

I was under the impression that the higher-ups in the Pentagon (Houseman) didn't want anyone to have knowledge about Shadow Moses, so Snake was also programmed to be killed by FOXDIE after he infected everyone else*, but then Naomi edited the program to a wildcard value to make Snake suffer more psychologically as part of her revenge.

Then again, in MGS2 Liquid says something to the effect of "no wonder you were passed over for the FOXDIE program", which makes it sound like Snake was only a carrier for FOXDIE...

* Naomi: "Listen, Snake. I'm not the one who made the decision to use FoxDie."

and

Campbell: "I'll bet the boys at the DIA and the NSA never expected you to come home alive."
Snake: "Me neither. I better not show my face around here."

Wait, didn't Naomi set it so that it will one day randomly kill Snake?

Argh, so confused now. So was Snake supposed to die or not and did Naomi edit it to kill Snake or to randomly kill Snake. Thanks.

Isn't Foxdie random to begin with?

Okay, after doing some thinking, I get it, I'm almost 100% sure I'm right now.

FoxDie was injected into Snake to kill Liquid, the other FoxHound members, Baker, the Genome Army, etc. Snake was to carry the virus and it would spread to all of them, killing them. Because the virus was meant to kill Liquid, this must mean that Houseman knew it would kill Snake too. He obviously knew they have the same genes if Snake was created out of the Les Enfantes Terribles project along with Liquid (And Solidus). Naomi, for some reason, edited it so that she could feel as though she got revenge, and that Snake wasn't just a tool of the government sent to die, but Snake was dying at her hands, and she would soon reveal it to him. I think that's right!

how come revolver ocelot does not die as he has liquids genes in his hand

FoxDie needs to be triggered in order to kill someone who is infected with it.

Also the nanomachines and vitals from Liquid's hand aren't that many so they could transfer a significant amount of the FoxDie virus into Ocelot's body so it could kill him.
Even if they had... Liquid Ocelot seizes control of SOP in MGS4 so he can apparently control the virus too.