You are correct. The CAT interface is only used to send and receive control commands between the PC and the rig.
Audio is handled like you think. A cable from the rig's audio out to the PC sound cards audio in. To transmit in the digital modes, you need a way to get the PC sound card output into the rigs audio input, AND key the rig.
There are many ways to handle those duties. You can:
Place your MIC in front of the PC speakers. You can see the limitations here. Ambient audio getting transmitted, pain in the butt and inconvenient as all get out to do, etc...
Run the speaker output from the PC to a connector (whether the front MIC or a rear accessory connector) and use VOX. Biggest issue here is controlling the audio output so you don't distort or overdrive the audio input to the rig. There is also going to be an impedance mismatch and a possibility of ground loops. Another problem is if you use Windows sounds on the PC. Those could get transmitted as well.
Buy or make an isolation system where audio transformers are used between both audio lines as well as some form of volume control for the mic side.
For any of the above, it's also possible to take a signal off a Serial Port or Parallel Port to key the rig.
Buy an external sound card system like the SignaLink that eliminates all the issues above.
I mention the SignaLink only because I use one. It is a slick solution.
HRD is some wonderful software. There's many tricks and other uses that most people don't know about. If you have a rotor, for example, and it supports connection to a PC, then it's very easy to slew your rotor over to a contact with a click of the mouse button (I just found this one earlier this week).
Also, get ahold of the excellent HRD Utilities to handle LoTW duties. DOwnload it here:http://wd5eae.org/Software.html
Hope this has been helpful. If you'd like to contact me for more help, my private E-Mail is my call at arrl.net.
73 - Jim - KB1NXE