You could definitely override either widgets/mini-calendar-widget.php or else widgets/mini-calendar/list.php and add a link in there (please see our Themer’s Guide to learn about the template override process).
Regarding your second question there’s an example of setting month view to a specific month/year in this thread.
If you need it to dynamically move to whatever the next month to contain events is, rather than specifying a month and year, you could combine this with / modify the solution for the mini calendar suggested here (which may be the thread you already viewed).
However, if you need further help on that end please create a new thread and we’ll concentrate on the initial issue in this one. Thanks!