The answer to your question is "yes"; as long as within your application you can abide by the limits of the USB serial speed and such (and the limitations of the Arduino hardware, as Paul noted).
If you need a speedier interface (faster than USB serial), there are other options available there, too (Ethernet or WiFi, Bluetooth). Most applications do fine with USB serial.
You can set up the Arduino to control many things, but if you need a fully embedded solution, then develop the application using the Arduino, then set up a "standalone" version of the Arduino on your custom PCB as the "embedded" version (unless this is a "one-off" standalone application, where you're not going to need to make multiples of the device - then just embed the entire Arduino if your hardware enclosure has the space).
