Chapter 4 and Chapter 5 on RasPi set up both the home assistant satellite and wakeword as separate service running at the same time in the background, so yes the computer can do other things as well. As long as the total of tasks that are running do not cause a bottleneck in CPU, I/o, or network.
There have been conversations about using the same RasPi as media player and voice assistant satellite … and one day I will look into doing that myself. In one way it would help to have both audio out and audio in on the same device - Echo cancellation subtracts the sound it is playing from what the mic is receiving, and the difference is what the human is speaking - but I understand there’s more work to get it to alexa/google quality. Also, if I set up my voice satellite in the kitchen to play music for my spouse it will have to be good quality opera sound 