Hi, have you checked the forum, solidworks knowledgebase and the help? There were recently some really good conversations about how it works.
Self alignment basically means that it will rotate about the middle of where you've applied the bearing connector. Like a self-aligning bearing. If you don't choose that option, it will act more like a fixed hinge.
Stabilize shaft is a subset of the soft spring command for the bearing connector specifically. It prevents rotation. But if you have off-axis loads, it is best to add a real restraint instead.