My collection of FPS games with coop play as an option is ( Team Fortress 2 , Unreal Tournament 99 GOTY Edition, Counter Strike, and Alien Arena ). These games aren't the newest games out there, but are fun.
Alien Arena is free and if you against other players your better off vs up against the AI's in the game as for the AI's cheat. If you hide around a door for someone or some alien to enter the room to shoot it in the back, it is unfair in its code that it sees in 360 degrees when entering a room so it turns your way instantly shooting at you. When up against real opponents they wouldnt turn right away to shoot at you like that unless your location was tipped off to them by a teammate. The gameplay against real people is hardcore, so you have to be fast! If your looking for an adrenalin rush and like games that are fast paced this is fast paced. I had better gameplay on the move vs waiting to snipe people.
My all time favorite FPS is Unreal Tournament ( UT99 is my most favorite but graphics are dated ) but there is map editors and people hosting LOTS of custom servers out there to connect to for death matches capture the flags and coop play. About 10 years ago I use to host my own server for others to connect to. I had a map called Massive Bedrooms as the main map that ran. It was great big rooms that placed you and other players the size of army men and it was like a Toy Story deathmatch. So many great places to hide and snipe as well as ambush people. Climb into Lego houses and run across a bed for extra armor plating or ammo that would spawn because you can run out of ammo if you shoot foolishly to empty your guns. Carry as many guns and ammo as you can acquire and select fast with mouse wheel to which ever weapon you wanted as well as right and left mouse clicks were primary and alternate fire so weapons had where you could hold right click and stack up a RPG to shoot 5 at the same time or left click and fire one at a time. If you kept target in sight long enough you could get a missile launcher to lock onto a person and they had to run in a way to avoid it following them to their death. Additionally there is one weapon in the game that is the mother of all weapons. Its just about a nuke. You can lunch it or alternately guide it with missile view. And it doesnt matter if your on top of the target or a foot ball field away from them and other people the shock wave is instant death to any one in the blast radius of it. All sorts of futuristic and some modern weaponry. My server died when the CPU fan seized up on a hot summer day in a home with no AC going and that was the end of serving it up. It was just a weak Pentium III 850Mhz computer with 256MB RAM running Windows XP and hosting the game on a 12GB HDD. The best thing about this game is that it was created around the Pentium III computer era and so you dont need a powerful computer to play it. The game can be obtained cheaply. I bought my copy for $10 on amazon. The game does have some harsh language and wanton to it. If you snipe a headshot the persons head blows off and bounces in a bloody mess. I think there is a checkbox to remove some of the gore, but I never used that checkbox. You can stack kill types such as multiple headshots you will hear a mortal combat type of voice that yells out GODLIKE etc. There are also things about the game that I cant discuss here as for I dont want to promote misuse of the game. I still occasionally play it against others and sometimes just bots in which I will set player count to 16 so its me against 15 bots, then raise the kills to like 50 and go out and have fun.
Team Fortress 2 I played a little bit of, but due to a computer at the time that only met the system requirements of the game when it came out, others with gaming rigs and fast frame rates had advantage over me with 15 to 20 fps and so I havent played that much. These days I have a system that would play it well, but dont have any desire to play it. I have thousands of games some of which have rarely been played and others have been played to death and I keep coming back for more gameplay with them.
Counterstrike I played online against others until people started using cheats with scripts to instant targetlock, walk in the air and snipe you which one guy did, and jump spam scripts to avoid headshots. The cheaters killed CS for me. Not sure if they got better at keeping the cheaters out of CS, but I dont like playing games and getting killed with unfair advantage that cheaters have.