Event management problems
The calendar is not part of core CiviCRM, so I can't be 100% sure without looking at it, but the first issue may be that you didn't mark the event as "public".  If I were setting up a calendar like that, I'd want to limit it to events marked "public" in case I wanted to record other events that shouldn't go on the calendar.

Your other two issues could be so many different things that it isn't worth hazarding a guess without more information or access to your site.  If you are in touch with the company who built your site, you should contact them, or you might contact a CiviCRM Partner.