Sure! Bring the kids! We want everyone to have a good time, so it's up to you if you want to bring the little ones along (but be aware- alcohol and fire will both be present).
Casual. Come as you are, we don't care what you wear. Plan on being outside, in the summer, and possibly walking in the grass.
If your invitation says "and guest," then yes. If not, at least ask first (if we've met your date before, the answer is probably yes).