By default, all panels in game mode are hidden to ensure the focus on the gameplay. In the game mode, you cannot move and/or delete objects, nor edit animations.
For show left and right panels hit
CMD + / for macOS or
CTRL + / for Windows. Also you can go to
Main Menu >
View > uncheck
Hide UI. Note, that some changes you make in the Inspector will be automatically saved when switching to a different mode. It work for Main Character Properties; Main Camera; Gravity.