Nowadays, almost every audio interface has a driver with additional options that either expand upon or work separately from your built in audio driver so it will vary from product to product so you would have to check with the ones your interested in.
My Akai EIE Pro, for example, when used in Ableton can be set to have it specifically use any of the inputs and outputs or all of them. Even further, it lets me specify what inputs and outputs to use on each individual channel. Each DAW is different too but as far as I know, every DAW exports audio without any interference from outside audio sources (running applications, computer sounds, games, etc) making the point of whether or not your audio interface plays these sounds irrelevant. It would be a terrible design flaw if this weren't the case. Of course the only exception would be if you are using ReWire with an external 3rd party application. I do this occasionally with Ableton into Reason or vice versa since Reason doesn't support VSTs or other types of 3rd party plugins.
It also depends on if you have a Mac or PC. Macs have a utility in the applications folder called Audio Midi Setup (Applications/Utilities/Audio Midi Setup) which allows you to directly control your input and output channels. I have mine setup to play my computer sounds made by the computer itself through the internal speakers while having all other audio play through my monitors plugged into my EIE Pro.
If you were on a PC, it would all depend on your driver. The built in audio driver (which is not very good) has quite limited options on what lines in and out can be used since it only has one major input and output source. It is more than likely that your specific audio interface will come with its own driver to create additional options for the inputs and outputs. You might even have a 3rd party audio driver which already has these options. For example if you have the ASIO4ALL universal audio driver, it automatically is set to only play sounds from the application you currently have open and in the forefront. Other drivers have this feature too, so look into that as well.
I know back when I had a (very) old version of Realtek High Definition Audio Driver, it was a HUGE pain to get it to work with my MicroKorg. Even after purchasing an extremely simple midi-interface to try to circumvent using the audio input option, it was still a huge pain in the butt getting it to work and wouldn't even work consistently whenever I tried to use FL Studio. It could have been the driver, FL Studio, or even both since this was about 5 years ago and the software wasn't even close to as good back then.
I hope this points you in the right direction.