There's a dedicated scene mode for "Pets", but I've never used it and I'm not clear if its purpose is to freeze fast-moving subjects. The example image in the manual appears to be a sleeping cat, so you might be better off with "Sports Action".
Because I'm never sure what some of the RX100 scene modes are trying to achieve, I tend to do my own thing in these situations. I would probably set the drive mode to "Speed Priority Continuous", which will allow you to record a burst of images in rapid succession and to hopefully catch the running dog nicely in at least some of them. Beware that the camera will only focus once for the first image, so that won't help much if the dog is running towards you. I have "auto review" turned off too so that the rear screen doesn't keep locking up to show me what I've just recorded.
In low light, I might think about manually setting a higher ISO like 1600 (that's usually my limit on the RX100) and selecting the fastest possible speed in Shutter Priority mode.
I don't think it makes any difference whether you're shooting RAW or jpeg when it comes to scene modes.
Just to be clear, ISO 1600 is my personal upper limit for shooting in low light. On a sunny day outdoors I'd just choose a fast shutter speed in shutter priority mode and let the camera make its own mind up about everything else. The image quality is better at lower ISO.