Is it possible to play this game (and enjoy it to some degree!) without playing missions beyond, say, 1 or 2?
I tend to prefer sandbox games myself, where the enjoyment is derived from player-driven content and pvp. I'm fairly early into this game, but so far it seems to be a pretty linear series of missions and not much else*. Hopefully I'm wrong.. or maybe it's something that will change in time.
* Self-flagged as potentially ignorant statement.
Edit: I've read a little about Foundry (player made?) missions and fleets. What can player fleets do specifically (other than missions)?
In short, yes. It is possible to enjoy the game doing exactly what you're saying, and to play it, and level up.
However, it comes at a cost, since the game wasn't outwardly designed to be played like that. You'll level more slowly, playing UGC Foundry Missions, attempting (mostly in vain) to get PvP matches, and playing the Duty Officer System very heavily. There are some simple repeatable missions you could do to speed this process up, but, by in large, you'd be denying yourself a very large part of the game.
But, if playing like that makes you happy, you're willing to take that hit to EXP and stuff, and you honestly enjoy it, more power to you, friend. =D