Carrying on Shabbos and Yom Kippur in Crown Heights
There is no eruv in Crown Heights that may be relied upon. Carrying through the neighborhood streets is therefore forbidden on Shabbos and Yom Kippur, whether the object is a stroller, a key, a siddur, food, or something left unnoticed in a pocket.
The melacha of hotza'ah
Hotza'ah includes transferring an object from one halachic domain to another and carrying four amos in a reshus harabim. It applies to small and ordinary objects no less than to something valuable.
The status of the place, not the ownership or purpose of the object, determines the basic prohibition. A privately owned walkway may have the halachic status of a public area, while a properly enclosed area used by many people may be a reshus hayachid.
This is why the question cannot be reduced to whether a person remains near home or whether the object is needed for a mitzvah. The building entrance, front steps, sidewalk, street, and destination must be considered according to hilchos hotza'ah.
Everyday carrying
A key, siddur, tallis, food container, bag, coat item, or children's supply may not be carried through the streets in reliance on the publicized eruv. A stroller may not be pushed through the neighborhood for the same reason.
There is no general heter to arrange ordinary carrying through a child or a non-Jew. The details of chinuch and amirah l'akum matter, but neither becomes a standing weekly arrangement for carrying items that an adult may not carry.
Recurring needs are usually best handled before Shabbos: leave a siddur or tallis at shul, place food and children's supplies at the destination, empty pockets and stroller compartments, and arrange a suitable method of building access.
Medical, safety, and mobility needs
Pikuach nefesh and genuine safety needs are governed by their own halachos and do not depend on the existence of an eruv. When action is needed to protect life, it is taken without delay.
Non-emergency illness, disability, and mobility questions can involve the person, the device being used, attached objects, an aide, and the domains along the route. Recurring needs should be addressed in advance with a Rav who can consider the actual circumstances.