Shooting kids and pets are not the easiest, but check out How to keep flash from disrupting the scene?. There is a good discussion about using off-camera flash in a way that won't tick people off. Basically, what you want to do is control the color temperature of the light reaching the subject. That typically means use a flash that is more powerful than the ambient light.
If you are using flash, it's very good to set the camera on manual, and pick a shutter speed (say, 1/200) and aperture (say, f/5.6), then let ETTL on the flash adjust its output to make that exposure correct. There comes a time when the flash isn't big enough for just any arbitrary exposure, so be watching for relatively good results as you continue to shoot. Note: at a 200th, the ambient light should no longer be a factor; just the flash. I chose f/5.6 because it is a flattering aperture for most lenses in people shots and doesn't rely on razor thin autofocus accuracy.
To answer your other question: Kinda. Shooting RAW gives you a ton of options in post that would not otherwise be easy. Still, you will have mixed light and sometimes that can't be easily corrected. It's also additional work for each frame.
As a final note: Fast lenses have advantages and disadvantages. The advantage is you let more light in (Yay!). The disadvantage is that when you do open up and let more light in, you give up depth of field, so getting perfect focus is crucial. I can tell you stories about great shots I almost got where the camera's AF picked the tip of the nose instead of the eyes and the depth of field was that shallow: from the tip of the nose to the eyes, so the eyes were not tack sharp. So be forewarned, a faster lens is not a sliver bullet but rather another tool you can use to good effect.
