How can I have teachers add site events?
Make use of a Calendar editor role and assign the teachers to that role on the site.
Why do events without duration not show on the first day of every month?
If your timezone settings are not default you may find that events entered with no duration on the first day of the month, do not show in the month view of the calendar. Setting the two timezone settings in Administration > Location > Location Settings to the default settings should resolve this.
How can I show more upcoming events?
To show more events in the Upcoming Events block, increase the days and/or events to lookahead in Site Administration > Appearance > Calendar.
Can I use a different calendar type instead of Gregorian?
Yes. See Calendar types in the Developer docs. It's possible to set the calendar type preference on the user and course settings page, the same as the language preference. However, unlike the language select box, this option is only shown if there is more than one calendar type. The reason for this is because the majority of sites using Moodle will not want to use another calendar type other than Gregorian.
How can I import calendar events?
See Calendar import.
Can I have a two-way sync with my Moodle calendar with my Google calendar?
See this forum post: How to sync Google calendar with Moodle calendar
Why are month names not being displayed in the correct language?
Names of days and months are pulled out of your operating system. The system is configured for Moodle in langconfig.php where you have the strings locale (for -nix type operating systems) and localewin (for Windows operating systems) that should point to the right locale on your server. Your server operating system should support the language (= locale should be installed). See MDL-31622 for further details.