Welcome to the site, you'll be helping others in no time, trust me!
1. to connect your Axim through a bluetooth enabled phone read there:
http://www.aximsite.com/bluetooth/
to connect your Axim through wireless network read here:
http://wifi.aximsite.com/
co connect your Axim trough your leptop open Active Sync > options > pass through > select internet
2. reinstall the software, update it(google for ActiveSync 4.1), reconnect and re-establish the partnership.
3. Yes, Axim site - the best free and friendly help you'll ever get on your Axim
4.Have no idea - I use Orange plan
also read the first 3 links in my signature
Welcome to the community, you''l like it here.