When you're connected to your office computer over the VPN, your office computer is the primary computer; your remote computer essential becomes just another peripheral for the office computer. So printing capabilities are restricted to your office computer while you're connected. To print documents stored on your office computer locally, you'll need to download those documents to your local computer and print them after you've disconnected from the VPN.
No. In fact, your first choice for remote connectivity should always be to use Secure FTP (File Transfer Protocol) to connect to your network folder. That approach is much simpler and ties directly into HSU's network access control system, so it's just as secure but much simpler and more reliable to use. The VPN system is really designed to be used only by those people who are already connecting to their office computers remotely using Remote Desktop or VNC.
A Virtual Private Network (VPN) is essentially a secure communications "tunnel" that connects a remote computer to a corporate (or in HSU's case University) network. HSU's VPN uses a combination of authentication (confirming that you are who you say you are) and encryption to ensure only authorized users can read data transmitted through that tunnel.