Make a connection to it from Outlook with the Outlook Connector or use
Windows Live Mail from the Windows Live Essentials Suite. Then the calendar
will be cached to your machine and changes will be synched when you get
on-line again.
Yes, you can connect with Outlook 2003.
I'm not sure what you mean with "Windows Live"; that's just a brand name
where multiple products fall under. Do you perhaps mean Windows Live Mail?
Yes, that should work as well.
Which sync errors do you get when?
What does the log say?