Can you recommend me modern war FPS games? (read description please)

Far cry 3 is a pretty damn good modern shooter.

Spec Ops: The Line or Far Cry 2 or 3 would be my recommendations for 'modern, non-fantastical' shooters.
They're basically the only ones I've ever had interest in, as I stay away from [non-co-opable] world war/modern warfare shooters [without creative license involved in the content] as a rule.

bf hardline :)

Play GTA V as first person

Spec Ops: The Line, though it's third person and it's focused more on the story than gameplay.
You don't really have to kill animals in Far Cry 2, though it's much more boring than 3 and I couldn't finish it.

Metro?

Squad

Tried Sniper 2: Ghost Warrior?

Rainbow Six 3 (Awesome game, and I could not help but feel deeply disappointed by later installments (Lockdown & Vegas))
Ghost Recon: Future Soldier (Would have been an alright game, had the AI not been utterly unable to deal with how I play games, and of course bugs (It's a ubisoft game after all)).
Slightly older but the Delta Force series is always an option.

The first two Soldier of Fortune games, were a lot of fun. But they're old, and impossible to find (legally speaking).

Modern war, Singleplayer campaign?
CoD 6: Modern Warfare 2, CoD 8: Modern Warfare 3. That's it.
You've played others already, and these two are the last bits. Oh and maybe Black Ops 2 I guess.
Maybe CoD games are horribly short, but there isn't anything else if you're looking for something at least decent.

CSGO, Insurgency ?

in terms of mechanics and gameplay, Red Orchestra 2 is the best fps I've ever played, RO Vietnam is coming out soon

SOCOM 4: U.S. Navy SEALs (exclusive for PS3) but it is TPS
Project I.G.I.-2: Covert Strike (can be found on gog)

You have awful taste in games... All those games you listed are just as bad (in the case of Medal of Honor Warfighter even worse) as CoD. The ONLY military shooter that's set in modern times that I could ever recommend is Spec Ops The Line. But since it's a third person shooter, it's not what you asked for. But seriously, it's the ONLY GOOD one. And the campaign actually has a respectable length.

I enjoyed Call of Duty: Black Ops (the first one) a lot, because it has a lot of historical elements but it's not focused on one thing (like a war) in particular. It has several missions in very different areas and contexts, following an interesting story from different characters' points of view.
There are many weapons with several attachments, and some missions have 'modern' mechanics, like calling helicopter support, shooting a guided rocket, supporting a land team from an airplane...