First, you need to make sure that the widget is active; through this link. Then, follow this link to make sure booking is active and working. Finally, follow this link to make sure that the event in question has at least one ticket. Then the button can be seen.
If you like to use the widgets like this Page https://webnus.net/modern-events-calendar/widget-view in the demo, you need to create the shortcode and then go to WordPress Dashboard > appearance > Sidebar and add the MEC Single Sidebar widget in the sidebar of your choice. Then find the shortcode in the dropdown list and use it. Note that in order to have that carousel you can only select the Grid type. It will automatically change to the carousel.
Yes. You can do this.
Let me know if you have any other questions in the comment area.