Funnily enough, (1) is one of the first things I wanted to do.
You’ll need to create an override for your index.hbs as described here:
https://getpublii.com/dev/theme-overrides/
And delete this part in the copy:
{{ translate 'common.updated' }}
<time datetime="{{date @website.lastUpdate 'YYYY-MM-DDTHH:mm'}}">
{{#checkIf @config.custom.formatDate '!=' 'custom'}}
{{date @website.lastUpdate @config.custom.formatDate}}
{{else}}
{{date @website.lastUpdate @config.custom.formatDateCustom}}
{{/checkIf}}
</time>