Could I read USB data from the mini USB port on an Arduino board, or is that only for installing Sketches?
What do you mean by "read USB data"? If you are referring to devices like keyboards and barcode readers, then no. You'd have to have a board with "USB host" capabilities. That said, apparently there are shields that can provide this functionality.
You can read data from your Arduino, using a program like hyper-terminal on windows. To do that set up hyper-terminal to listen to which ever com port that the Arduino is on with the correct baud rate (Same as in your sketch)
The Arduino is a simple USB to Serial device as Aoi Karasu said.
If you are on Linux or OSX, you can use the command:
screen /dev/ttyACM0 9600
ttyACM0 is the name of the device on my system, you may change it according to your system.
9600 is the baud rate, again you can change it.
Arduino is not a genuine USB device. It acts as a Serial to USB adapter. You have to scan all COM ports in the system to find which one is it and then talk to Arduino (perform read/write operations) using standard serial port communications that come with your development environment.