I just started the game, and by default it looked as if it were pressing me to play co-op right out of the gate. Is the game supposed to be played co-op? Will I be missing anything if I play through single player (besides the obvious joys of killing zombies with friends)?
You can complete the entire game and miss nothing if you play solo. The only reason to play with friends is, like you said, the enjoyment of killing zombies with friends. Frankly, I've found that playing with others makes the game way too easy as you can just kick + curb stomp your way to victory against most zombies.
Playing in Co-op only ups the difficulty and increases rewards gained. Other than that the story and gameplay is the same.