The game doesn't need to be on the SSD because you really won't get much benefit overall. Games use CPU/GPU/RAM/Internet more than they use your disk. The disk will really only be used when loading new content into memory.
With that said, your FPS on your game will not really change with a SSD vs HDD, since the HDD is not the bottleneck. For recording video, you should always consider using a drive that isn't being used by anything else. So if your OS and games are on Drive A, then use Drive B for recording your video.
One last important point is that for most cases, your HDD isn't the bottleneck when recording HD video, that is why people still use HDD's for their recordings. SSD drives are great, but you have to remember that the speed of the drive only matters for reading and writing. When recording video, your GPU/CPU/RAM are used first to process the video and then the disk at the end to save it; this all happens in real-time but don't forget the two steps.