Can I use the Calendar Server on OS X Server to sync Calendar

It is possible to set up OS X Server with services for calendar and contacts for the users, and when you use the same user login on different devices, the contacts and calendars will update across them (assuming you have network connections to the server and the devices are set to automatically get changes with push notifications). The only issue I've seen with this is when you set up the services on a Mac, you will have the "On My Mac" calendar and contacts group (iCloud does not show this when you set up a device with iCloud as it really is modifying the local copies of the data and pushing changes to the server).

It is possible to connect to shared folders hosted by OS X Server if the server and network are set up correctly. To access it over the internet, you have two options:

1. Port forward (possibly with NAT translation) AFP or SMB ports through your firewall to your server. This would allow you (from a machine on the internet not on your local network) to access the shared folder in Finder as afp://yourserver.example.com (where yourserver.example.com is your domain name, or in the case of a home server, a dynamic DNS name that updates when your external IP address changes).
2. Setup VPN on the server and allow VPN services through your firewall. By creating the VPN, you would have access to your network (the one the server is on) like you were locally attached to it and not over the internet (with the exception of Bonjour services. Bonjour broadcasts don't work over VPN). The VPN (if configured properly) would add another level of encryption to the data transfers (I don't recall off the top of my mind if AFP is encrypted by default). You can also set up DHCP and DNS with a primary zone and add the name of your server (and/or other computers) to this zone and be able to use Bonjour like addresses (like myserver.local).

As to which is 'better,' I would recommend the VPN route as I am cautious opening ports on my router/firewall and would allow access to more than just the one machine. But in the end it will depend on what you feel like trying (with the help of these forums, other forums, and Google) and how paranoid you are about security. If you implement the VPN, you could access your shared folders, your contacts and calendars securely from the internet, but you would have to have the VPN enabled to get push notifications/updates.