How can I make Intellisense show up the function parameters once the parentheses are written down? It shows only if I just type them like in the tutorial example, but not once they are already written and I set the cursor with the mouse inside.
I think you want the Trigger Parameter Hints command. If you type it in the command palette it will show your current short-cut (mine is Ctrl+Shift+Space).
You can also see (and optionally change) the short-cut in File/ Preferences/ Keyboard Shortcut: