I wouldn't have a problem with a 13 year old playing CoD... Yes it's obviously more mature then say Fifa or Star Wars, but the whole "Blood, guts and gore" is widely over hyped and not actually the main point in the game.
It's more like gaming/cartoon war, just shoot someone quickly in like 2 seconds and then run off, capture a flag, toss a smoke grenade over a wall, etc. It's not like torturing prisoners or massacring civilians. I think you can actually turn the blood off in the settings, and on some of the more "hardcore" missions it gives the player a choice to skip the mission (With no penalty to the player).
As for online play, if your child has no mic then they cannot talk to people, simple as, even if they do have a mic, you can change the settings so other players voices come out of the TV speakers, so you can hear what's being said by them, and your child in the room (Who you can obviously hear if you listen from in the room or out side of the door)... But yes like everything public, there is sometimes some mature language used, but you can block other players so you cannot hear them and so they cannot message you.
In my experience 90% of 18+ games are fine from around 13/14+, the same with 16+ games from say 11/12+ and pretty much no bother with 3/7/12+ games. But obviously at the end of the day it's your choice and you know your child the best.