In Ventura (13.2), you can turn off automatic updates in the “General” tab, then “Software Update” and click on the little “(i)” icon to the right of “Automatic updates”.

When I saw it the first time, it was unclear to me that I can even click on it, since it doesn’t look much like a button. They shouldn’t hide such important settings in a tiny icon or at least should have made it much more apparent that it is clickable.

Couldn’t find any settings related to the closed lid though.

It's actually "Wake for network access" setting, see here: https://youtu.be/OHKKcd3sx2c?t=349

Thanks for pointing to that. I don't think I've ever checked this setting, but I just did and in the latest macOS it's located in System Settings>Battery>Options>Wake for network access>Only on Power Adapter and it was already set so I'm good. Hmmm. I don't remember ever changing it from the default.