Will the Nikon 1 J3 be good for indoor sports? I don't want a full size camera and want to keep the cost below $800.00. I am torn between getting the Nikon J3 with the all in one lens, or the kit with the two lenses.
Unfortunately, no. Since all the kit lenses are slow/dark, they aren't well suited for indoor sports. And the only really fast/bright 1 series lens is just 18.5mm, which even with the 2.7X CF is only 50mm (full frame equivalent), so probably not long enough for gymnastics, especially floor. Although it might be OK for events like vault if you can position yourself right next to the apparatus. (And even if you can, your daughter may find that distracting.)
Also unfortunately, indoor sports are the hardest thing to capture on a budget. (There's a reason the pros who shoot NBA games, for example, use equipment that costs several thousand dollars.) My best recommendation for something small that could do OK would be an Olympus E-PL5 (~$600) + Olympus 75mm f1.8 (~$900) which is obviously twice as much as you want to spend.
Full disclosure: I own an E-PL5 and 75mm lens - Mark