I'm curious if I can use my existing EV3 and/or PowerFunctions motors to extend the functionality of my LEGO BOOST set. Is this possible?
While LEGO BOOST is compatible with the bricks from other sets, the motor, distance sensor, and Move Hub are not electrically compatible with current EV3 or PowerFunctions components. BOOST uses the same six pin connector as WeDo 2.0 from the Education line:
So, the bad news is that you can't easily use PF or EV3 motors with BOOST. However, LEGO plans to use this connector for PowerFunctions in the future and possibly for the successor to EV3 as well, so there may be more compatible components down the line. Here's how it's explained in the WeDo 2.0 FAQs:
Is this a new plug system?
Yes, this is the new LEGO Power Functions plug that has been optimized also to meet potential future needs.
What does that mean for the existing plug systems on other Power Function and MINDSTORMS products? Will they also be changed?
Yes, eventually we will convert to the new plug system after a transition period. The exact timing of this transition has not been determined.
While the Boost components use a different connectivity, there is potentially another way you can extend your EV3 set with the capabilities of Boost by having both communicate via Bluetooth.
Unfortunately, there are no standard LEGO-provided tools to do so, but the systems are open-ended enough that some people are working on it. For example, here is a Python library which you can use on your EV3 brick running ev3dev (a specific debian distribution for EV3) and talk with the Boost hub. I haven't tested it myself but it looks promising.