@vlz-iris -- I looks like no one answered your questions, so I will take a stab at it:
1. Using the Microsoft Project desktop application offers WAY MORE functionality for editing and managing an enterprise project. Managing a project using the Project Web App user interface offers you a greatly reduced feature set compared to Microsoft Project. To use Microsoft Project with Project Online, however, keep in mind that you need to create a Login account in Microsoft Project and set it up to connect to your Project Online instance. This is what allows you to open, check out, edit, save, publish, close, and check in enterprise projects in Microsoft Project.
In response to the second half of your first question, no, only one PM at a time can open an enterprise project Read/Write. Project Online does NOT offer co-editing of projects.
2. You can control access to projects using the security Groups and Categories. However, for that to work, you should not use the Team resource feature for your sub-managers. Simply add them to the project team and then configure default or custom Groups and Categories to control user access.
3. The Team Planner view is available only in the Microsoft Project desktop application. There is no Team Planner view, or any view even close to it, in Project Web App.