A. It is, and you’re right, we’re allowed to carry in private property on Shabbat. But in a Jewish legal sense, open, unfenced area, while not considered public area, is still not private for the laws of carrying on Shabbat without an Eruv-string or fence.
Some active Jewish communities, indeed whole cities or towns, have established an Eruv to enable carrying within on Shabbat. These rules are complex and detailed with a variety of Halachic opinions. We once did a Torah-Tues class titled “The Eruv Controversy”.
FYI: There is no Eruv around UAlbany or the nearby surrounding areas, while there is an Eruv in the local Albany Jewish community that runs east of Route 85 and south of Western and Madison.