(they're not "perfect", but I can't spend forever on each CG or else I'll never finish)
Yes, you will be able to name the player character, "Ai" whatever you want.
Yes, there is more of a back story to "Ai". Personality wise, she's somewhat of a loner in this game. "Introverted" is probably the best description.
Yes, there will be holiday events. Examples: Valentine's Day, Christmas, etc...
More info on this here