Edit Top 200 Rows is simply a GUI way of performing INSERT, UPDATE or DELETE operations on a table, so you only require the same permissions you would require for performing these actions via T-SQL scripts.
The minimum required is the following:
Contained Database User or SQL Server Login mapped to Database User
VIEW ANY DATABASE on server to Login
The Object Explorer pane and the Query pane are independent. You can have the Object Explorer connected without having a Query connected, and vice versa.
The menu option you probably want is Query -> Connection -> Disconnect.