Best DSLR + lens combo for low light, indoor kid shots

Don't get too hung up on the light loss; it's not of any real practical significance.  If you're shooting in auto, aperture priority or any mode except shutter priority, the camera won't necessariy compensate for the light loss at the sensor by raising ISO and noise, it can just hold the shutter open longer (say 1/60 instead of 1/100 or something).  For most candid shots, this won't cause you any motion blur, and the fact the Sony has in body stabiization means it keeps camera shake under control with any lens, which is a big advantage over a Nikon when using a prime.  I take these sorts of pictures of my kids all the time with an A65 and don't have any problems with motion blur and noise is well controlled.

The only time I've had any difficultly has been at my kids' gymnastics practice, where the light is terrible, they're moving fast (bouncing on a trampoline, at a dead run flipping over bars, much faster motion than indoors at home), and where I'm seated fairly far away from them.  There, I'm too far away to use my 35/1.8 prime, and my tele lens is a 70-300 F/4-5.6, much less light gathering ability, so to keep motion blur under control I have to shoot at relatively high shutter speeds (1/250 or so) and then ISO goes up to 3200-6400.  Even then, the pictures look fine fit onto a computer screen or printed small; it's only blown up to 100% that they look noisy and you see the detail lost.  But shooting with a Nikon wouldn't improve it at all, I'd still be at ISO of 2400+; the fix there is a better lens, like a 70-200 F/2.8, which at 200 would give me 4x the light gathering, letting me drop the ISO down to 800 where noise isn't a problem.  So, you can see its the lens that makes the big difference for that sort of special situation, not the camera.  Getting a fast telephoto like that probably would help even more than going full frame, but those lenses run at least $1000, and I can't justify that right now.

So, pick the camera you feel most comfortable using without getting hung up on some of the technical minutae.  Either the Nikon or the Sony will serve you well; I wouldn't be overly concerned about the Nikon not focusing well either, it's probably the grumblings of somebody in a very challenging situation that would be tough no matter what the camera.  Then get a 35mm prime for the indoor stuff, and have fun.