You can enable/disable whatever blocks you believe are the best for use by the content manager. You can have all Cwicly blocks or all WordPress default blocks or a mixture of these and third-party blocks, it is completely your choice.
The only slight quirk to be aware of is that if a user is pasting formatted content (e.g. lists, etc), these will currently only work for the WordPress block. If they are primarily using the editor to create content then this is not an issue.
The benefit of using the Cwicly blocks is that the user has access to the advanced user interface that Cwicly provides in the inspector.
When you use global styles, global elements and global classes with relative styles on your containers and post-content blocks, all of these will apply to any block used (including the WordPress ones).
Please see this thread for discussion:
All of the bugs mentioned there have been resolved and both block styles and classes (with relative styles) all work correctly now.