The answer is yes. There are cables that allow you to do video out of the GoPro and use as a recorder and FPV camera at the same time. I was about to do this, but after some digging, decided it is best not to.
1) There is some lag. Upto 1 second, camera dependent. 1 sec many not seem like long, but when you come up on a tree branch, fence, etc, 1 second is a long time to be behind.
2) I've ready many reports of the video out randomly stops working, stops recording, etc. So, the money saved by using one camera is quickly spent replacing your craft.
3) The video out quality depends on your video transmitter and reciever, not the camera. A HD gopro will NOT give you a better picture than a quality FPV camera.
4) GoPro have a separate battery. One more thing to charge or forget to be charged, or go dead mid flight. There are cables that will allow you to charge the GoPro while in use via a 5v source, but once again, more complications.
Do some more research and make an intelligent choice.
For reference, I have not flown my quadcopter FPV yet. Still in a kit to be built. My advice may not be well validated.
Of the half dozen or so FPV quads I have I fly two of them exclusively through the GoPro, I have yet to see any kind of lag that would be problematic. If there is any appreciable lag it's well under a second as I've always managed to avoid tree branches and other immovable objects but maybe that has more to do with my 100 plus hours flight time behind the goggles.
Using the video out connection I have yet to have the video stop working due to anything related to the GoPro. I have had it stop working due to a broken wire between the GoPro and the VTX but that can happen regardless of the camera in use, that's when it's really nice to have that RTH switch on the TX and know that it works.
Video out quality does depend somewhat on the video link but also on the camera and the method you use to view the video. I've had three sets of Fatshark goggles so far, the Basic, the Dominators, and now the Attitude. By far the basics had the worst video resolution, it was a revelation to swap to the Dominators and see how good the video could be from the downlink. Unfortunately the Doms don't have adjustable interpupillary distance as the Basic and Attitude do so I have to sell them and move to the Attitude. In doing so I lost the ability to swap the receiver in the goggles, the Attitude video RX is built in and not behind a removable hatch as on the Dominator, but gained a slightly better resolution on the screens and a much stronger 5.8 receiver. There is a definite difference in color and image clarity between a GoPro @ 720p HD vs. a dedicated 540 line FPV camera when viewed through Fatshark goggles. The GoPro view is wider and sharper by far, making it much easier to see what is around you vs the more straight ahead view of the FPV cam. the one area that can be problematic is when the GoPro is pointed towards a bright light source such as flying into the sun, the video is washed out and detail disappears until you move the lens away. Not so much of a problem once you've got a lot of FPV stick time but a real issue for beginners and something to keep in mind when deciding on which camera to use on a first FPV multi.
As far as the GoPro battery is concerned, make sure to fully charge it BEFORE leaving for the field just as you would the battery for your TX or any other piece of equipment you depend on to fly. Beyond that, get a battery backpack and install it on the GoPro, effectively doubling the battery capacity to the poin t of being good for an entire day of FPV and video recording. If weight is a concern you can purchase extra battery packs and remote chargers so you can have a fully charged battery to swap over to if the current one gets too low. Check the battery status before flying and if it's at one bar change it out, they drop off extremely fast one they get to one bar and it's good practice to not fly at that level.
That said, you can spend a lot or a little on an FPV camera and expensive is not necessarily better. Lately I've been using the Fatshark RCV922 cameras and they've proven to be pretty darn good. There are better cameras in the same price range but the LHS doesn't carry those, there's something to be said for being able to just go to the shop and get the camera when I need it vs having to wait a week and paying shipping costs for ordering online.
Ken