I think you've answered your own questions so I'm just going to validate it. He can pick up your/his child anytime with reasonable notice if you're allowing it before there is a structured visitation schedule. If he wants to take a holiday picture, that's on his dime, but since he has given you nothing, I wouldn't help him arrange this in any way. Don't take her to the photo place, don't research it, and don't take her to the nephew's birthday party with a gift.
Put your "permission" in writing either through an email or a letter to your ex. Just say that you are meeting her necessary expenses (food, clothing, diapers, formula) and that you cannot finance a fancy outfit or a photo session or any other luxuries. Period. I wouldn't put a whole lot of detail about how ticked off you are - just be business-like and then you will have proof that you offered visitation. Keep his responses in which he lists all the things he wants you to do on his behalf. Do not engage in endless phone calls where he tries to manipulate you - you will have no proof of what he said, and he will just continue to influence you. I would, if I were you, take notes and keep a diary of phone calls, with date and content.
You do not have to provide him with equipment like a stroller or a car seat - he can buy these at consignment shops or yard sales like everyone else. He can figure out her size, her diaper needs, and so on. In her interests, I would consider giving him your diaper bag with 1 diaper and the right formula so that she isn't hungry, but other than that, I wouldn't help him out much.
Yes, he will be in for a rude awakening. Just take the high road. It's not going to be fruitful to ask him to engage in school searches or doctor's appointments. He's not that kind of guy, at least not at this point. He's into the "public" displays of dedicating parenting - like photo memories he can show off to others. He's not into the day-to-day. Your best bet is to have those needs included in a support agreement so that her medical and education needs receive proper funding from him.
So don't give him any ideas, don't find a place, don't dress her in anything other than her daily clothes. Let him do it. Just say it's his idea and he should plan it. End of conversation. You do not need to stay on the phone with him about this stuff - just say that if he wants to talk about her needs, fine, but his needs/wants, you don't have time or the right to structure those! "Oh I wouldn't dream of telling you what to do. You'll have to decide what's best in that area."