Wayland, US
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Let's start out trusting P-mode in at least this: if it thought that 1/60 at f/5.6 at iso 400 would work, AND the images came out well exposed, that should certainly be your starting point, noise or no noise. If the 18-105 is like other Nikon zooms then the minimum aperture is reached in the vicinity of 50mm. Even with the VR, you need to worry about subject motion blur. You will need to exercise care if you shoot substantially slower than 1/60, so the only escape routes to higher shutter speed you have left are either the iso setting or a faster lens.
What you don't mention is whether the test shots you have taken represent the lighting that will be used. If there will be no theatrical lighting and you will be relying on the ambient gym lighting then, unless they have more light up their sleeve, what you've seen is likely to be what you'll get. If there will be some additional lighting and a corresponding dimming of the ambient, then things might be more hopeful but can get more challenging.
In any case, consider using center-weight or spot metering - especially with theatrical lighting. You don't care about the background, you want the dancers to be correctly exposed. If they are correctly exposed then they won't be noisy, it really is as simple as that. At 50 feet and f/5.6 your 105mm lens will have a depth of field of about 15 feet, so, as long as your focus point does not leave your daughter (or the dancer of current interest), you should only need to worry about blur. 1/60 sec. may seem a bit dicey for dancers but if you can time your shots to their holds at set poses, even that can work satisfactorily. See the first photo - D300, 50mm 1.4D, 1/15 sec., f/2.8, iso 1600 (not a typo - 1/15 sec.- notice the blur in the hands of the second dancer). Don't get too impressed by the camera's extra zero. Same sensor as a D90 but the D90 possesses the later, more competent implementation. With a D90 I have shot satisfactorily at iso 1600 and above. See the second photo - D90, 35mm 2D, 1/100 sec., f/3.2, iso 3200. Both these photos are uncropped and were taken at rehearsals which allowed me to roam around close in with a short, fast lens. Same church, a few years apart. Some relatively strong spots are on in photo 1, not in photo 2.
Noise starts to be a problem if the image either needs to be severely cropped or its exposure substantially boosted in post. At 50 feet an early teen standing erect should just about fit into a frame in landscape orientation shot with a focal length of 105 - even shorter if her arms are stretched over her head. This suggests that the tradeoff between the 18-105 and a faster, shorter prime might be close to a wash - with a short prime at 50 feet you'll be cropping more.
Best,
Hendrik
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