Talk through how you want to deal with this in your family, but here are some suggestions:
Give daycare, pediatrics, classmates parents, and family and friends your contact info and instruct them to call you first hand for appointments, meetings or emergencies. Don't ignore calls from babysitters, daycare or medical staff regardless of how important your current meeting is. You're the primary contact.
A lot of them will still probably call your wife first no matter how hard you enforce this, so enforcing this will split the burden and responsibility.
Make sure you know your child's SSN, birthday, allergies, sizes, current number of socks, favourite toys, and teachers and friends. Dress them for the weather they will be in throughout the day, and always pack extra because they will get wet or dirty or lose stuff.
Keep a family calendar where you can see appointments, and make sure to take a majority of those appointments if possible (plenty of responsibility will be pushed onto your wife regardless).
Add playmates and family members birthdays and other events to the calendar, and go shopping for gifts at least a few weeks before, without being prompted by anything other than the calendar. Wrap the gifts yourself or pay for wrapping at the store.
Let your wife know (or maybe add a checkbox to the calendar and check it off when the preparations are done) that it's taken care of if it's a shared task, but don't bother her with it if it's your specific job - that's your responsibility and nothing she needs to keep in mind at all.
Keep count of diapers and socks and formula etc, and buy new (in the right size) without being prompted. You don't need instructions for every little thing regarding your own child, you can take responsibility without being given it.
Hmm. Probably no one.
Small amounts are weird to need "no questions asked" and large amounts are too large to give away "no questions asked". Partly for me financially, but mostly because there is a not insignificant risk of overdose if you give out large sums of money to someone being cagey about why they need it.
If they can't tell me what a significant amount of money is for, it's probably because it's something I don't want my money to go to, so I'm not inclined to agree under those premises.
At the least I would need a very good reason for why I can't ask questions ("I'm preparing to vacation as a fugitive, and don't want you to have to testify"), but by then there is no longer no questions asked...