Analog of class view in Visual Studio Code

Yes, there is now a built-in Outline View in VS CODE

Yes, there is the workbench.action.gotoSymbol command also known as outlining. Its default shortcut is CTRL+Shift+O.

If you have an extension installed which supports this command for the current file type then a list of methods, classes and maybe other symbols for the current file will drop down which allows you to jump to their definitions.