WordPress does auto-update out of the box by default.
For core WordPress, yes it does. For example if you’ve WordPress 4.9.5 then you should be automatically updated to 4.9.6. It’s possible your host disabled that but most hosts do not. They hate being the host of sites that aren’t maintained. Or you may have a version that does not work.
It’s possible you fit under one of those categories. You can always manually updated.
Plugins and themes do not auto update by default. For that you either need to update some code or use a plugin. I use this one for that very reason.
https://www.ads-software.com/plugins/update-control/
Yes, it has not been updated in 2 years. It doesn’t need to as it still works and my plugins and themes are kept updated.