Does Business Manager allow to provide different levels of access for different resources?

Yes, security system allows to configure role-based access, providing capabilities to provide conditional access to different tables and individual fields. For example, you can allow access to translation job, but close access to prices and amounts in that translation job. Or, you can allow to view something, but not edit or delete those records. See security system documentation.

Can TBM track time taken to complete certain tasks?

This functionality will be added in future. Currently, it is possible to track time in jobs, but not in assignments. Since LSPs often pay translators (especially in-house translators) on time basis, not per-volume, this request became pretty frequent. We'll work on this and implement time recording in future versions.

Can Business Manager handle file versions (file updates during project)?

Currently, this is not possible. This was added to the roadmap.