Combat Arms.
I haven't played it in a long time but as long as Nexon hasn't fundamentally changed the entire thing then it should be what you're looking for.
It's essentially Nexon's rip-off of whatever Call of Duty game is out at the moment. It has a smaller (but still fun) version of national socialist zombies, some co-op bot battles, cool gamemodes (I
highly recommend "Spy Hunt." Players battle for intel items that are dropped when a player carrying one dies. Once a player gets all the intel, they become a super spy and have ultra heavy weapons and high health. It then essentially becomes a sudden death boss battle between the super spy and remaining players.), and a bunch of really nice maps.
There's a lot in the way of player customization, too. There's a digital marketplace with hundreds of guns, dozens of support items, different body armor, player appearance options, etc. The large majority of things can be bought with credits earned from playing through matches, whether you win or lose. There is a marketplace of items that can be bought with actual money, but those items are really not all that important.
