Which day of the week to be used as first day of week is implemented as (user/site) language dependent. If you always want Sunday or Monday regardless of the language used by a user, you would need to make two adjustments.

In model.php I would suggest to alter the return value of the function event_calendar_get_first_day_of_week() to return always either 0 (for Sunday) or 1 (for Monday). If you are not using the full calendar view, this change should be enough already.

For the first day of week in the full calendar you would need to modify mod/event_calendar/views/default/js/event_calendar/event_calendar_fullcalendar.js. In the init() function there the option firstDay: locale.firstDay currently used. You could change this to either 0 for Sunday or 1 for Monday regardless of language used and it should work.

Widgets without user icon: I guess with a change in mod/event_calendar/views/default/object/event_calendar.php you might be able to achieve this. Try setting $icon to an empty string additionally in the if-clause that checks for widget context

if (elgg_in_context('widgets')) {
        $metadata = '';
        $icon = '';
    } else {

Not sure though what the output will look like then. But I guess the icon column should be at least much smaller.

Thank you again! Both those suggestions worked like a charm - and the output without the user icon looks just how I hoped. I very much appreciate the time that you put in to make this all so easy for us.

My respect to you.