If I am only using one computer that will be hard-wired (with no other internet enabled devices), do I even need a router? Can I just plug it in directly to the modem (are there safety issues?). Thanks.
I'm not sure that's entirely true. There are risks from being connected directly to the modem that being behind a router (wired or wireless) avoids, such as port sniffing.
Hacking a wireless network is a potential vulnerability, too, but that's still not "in" your computer, unless you've got public shares.
When you say there are no Internet enabled devices, does that mean that there are other devices or that there are no other devices.
If there are other devices, what are you doing about DHCP ? Where are these other devices getting a local IP address ? If they are all USB attached to your sole device, then you are OK without DHCP as your device will get an IP address from your ISP and the USB devices don't need an IP address.