No
yes he can.
If she has full custody, then yes.
No, legally he cannot take them with him without your permission.
If there is a court order for custody, visitation or child support she will need his permission. See related question link.
They can not print this picture without permission.
no
No. However, if they are on your property without permission when they take the picture, that could be considered trespassing.
no
Legally, in most cases, yes. Ethically, no, not without permission.
What should a notarized permission letter to take a child out of the country with one parent say?
No, as both signatures are required on the passport.
no