As you note, first birthday is basically a party for you, not for her, except that she's the reason for the party. I would say that means invite whomever you want.
However, you probably should tend towards inviting people who have shown an interest in your daughter. Friends who have not shown an interest in her - at least, acknowledging her when they come over and play a bit - probably won't want to come to a child's birthday party, even when it's more of an adult thing, unless you make it entirely an adult party and don't invite her. Undoubtedly there are exceptions to this, but keep that in mind. I wouldn't say it has anything to do with whether they have kids - rather, whether they're interested in yours. Otherwise, have a party that's not kid-related.
As far as gifts, it's technically not polite to mention gifts on the invitation (as that presumes people will bring one). However, if you're worried about people bringing gifts, you can ameliorate that by changing the language some.
Come to a barbecue where we celebrate surviving our first year as parents.
That makes it less a birthday party and more just a get-together, so people won't feel as obligated to bring gifts.
You can of course also mention gifts if you want - while it's not technically polite, it's certainly not a major faux pas, and if you're worried about presents clogging up the works, mention away. I would change the language slightly, and not mention the alternative:
No gifts are required; your presence is all we need.
Then, if someone asks that they'd like to give something anyway, you can mention Unicef directly to them.