Can I see which staff member changed an order's status?

Yes. Open the order from Orders → Manage Orders and look at the Timeline.

Every status change is logged with a line like "Status changed from Processing to Cancelled — reason: customer". Under it you'll see the time and the email address of the person who made the change.

The same goes for shipping updates, edits, recorded payments and comments. Each one shows who did it. Entries Wcart adds by itself, such as confirmation emails, show "system" instead. Staff can't remove these entries, so the history stays complete.