That depends on several factors. First, is the other legal parent in the picture? If the other parent shares joint legal custody with your partner, chances are that person has priority for making medical decisions. Hower, the nonparent may be listed as an emergency contact on school and other important records, in case a legal parent isn't available to make decisions for the child.
Regarding school permission slips, nonparents face restrictions here. Schools are legally responsible for the children in their care. As such, they are permitted to accept signatures only from legal guardians. A legal guardian could be a foster parent or anyone who has legal custody of a child. An unmarried partner of a legal parent probably does not qualify.
Nonparents may be allowed to pick up a child from school (or day care) if a legal parent informs the school that this will be happening either regularly or on a particular occasion.
To maximize what a nonparent caretaker can do, the legal parent should contact school and other authorities to find out what rights are available and what procedures are required to establish them.