How to add a 2nd monitor to Macbook Air (mid-2012)
100% yes. Just keep in mind that you can't exceed  a certain resolution (I think it's 4K). Also, buy an hdmi dongle that is compatible (some cheap generic ones offer very poor sound and video quality)

You will be able to use as many apps as you'd like on both monitors simultaneously, and some programs (e.g. Photoshop) even allow you to spread them across multiple monitors.