The Admin interface is available at http://thermostat:7878/admin/. If you cannot access it at this address you can alternatively use http://smartappl.appspot.com/locate/serial_number from your LAN. Replace serial_number with the serial number of your thermostat.
This happens when you access the Admin interface at http://thermostat:7878/admin instead of http://thermostat:7878/admin/. This issues was fixed in newer models.
Calendar entries should have a 5 minutes duration. For example, if you want to send a cool75 command to the thermostat at 6 PM create a calendar entry between 6:00 - 6:05 PM with the subject cool75.
One of them (arbitrary chosen) is applied. Conflicting entries are not recommended. It is also not recommended to use the calendar assigned to a thermostat for other purposes. It makes sense though for a calendar to be shared by multiple thermostats and, thus, to be kept in sync.
Yes, see the API used by the iGoogle Gadget or the direct API.