When disabling automatic checking and updating, I expected the program to not make any external requests.

But it still makes requests to:

192.168.1.1 (my router) ; UDP 53
itch.io ; TCP 443
static.itch.io ; TCP 443

It would be nice if there was a way to disable this.

I think I fixed it :)

This:
fetchLatestVersion:function(){return window.fetch("https://itch.io/api/1/x/wharf/latest?target=billiam/deepdwn&channel_name=win64").then((function(e){return e.json()})).then((function(e){return e.latest.trim()})).catch((function(e){}))}

Replaced with this:
fetchLatestVersion:function(){return;}

When ‘check for updates’ is disabled, there shouldn’t be any external requests to Itch. It’s possible there may have been a request in progress when the setting was changed.

Can you try exiting Deepdwn after changing the Check for Updates setting and reopen to confirm that you’re seeing new requests?