Is every star formed in an open star cluster? As far as I understand, an open cluster is formed from a single molecular cloud, with each star in the cluster having roughly similar age and properties.

Our Sun is not part of any star cluster, but it could have been formed in one. Is every star formed in an open star cluster, or can they develop independently?
The question is still an open matter of current research.

It seems to be true that the vast majority of star formation takes place in groups and aggregates of various sizes - from a few stars to millions of stars in "super" star clusters.
This is likely because collapsing clouds of gas are normally much more massive than a star and the collapse process reduces the Jeans mass and renders the cloud unstable to fragmentation into smaller cloud cores.

However, it seems that the vast majority ($&gt;90$%) of star clusters/associations are either born in a gravitationally unbound state or become gravitationally unbound within a few million years. The gravitationally bound open clusters, who's exemplars include the Pleiades, are comparatively rare survivors (or at least partly survived) of this "infant mortality". So, in that sense we can say no, most stars are not born in open clusters, but the likelihood is that most were born in aggregates with near neighbours that went their separate ways shortly after birth.

Current thinking is that our Sun was born in a cluster of some ten thousand stars (Adams 2010). This is an argument based on the shaping of the early solar system by dynamical encounters and the early presence of radioactive nuclei that were likely injected by the explosion of a very nearby massive star (probably a cluster sibling).