I own the G7 myself, and I can tell you that I have no problem whatsoever in going up to 1600 ISO if that gives me a better exposure than 800 ISO. After a slight pass of denoising and sharpening in post, it's as good as if I had shot at 800 ISO. And 3200 ISO is absolutely usable as well, as long as you get a good exposure out of it.
I find the low-light performance of the G7 to be very good, just awful in the shadows. You can actually get less distracting noise by bumping to 3200 ISO and having a well exposed image than shooting at 1600 and having an under-exposed image. And if you get a good exposure, 3200 ISO is very usable after only a bit of denoising in post.
That said, your image is way underexposed, and at 6400 ISO that spells disaster. That weird halo might be the in-camera NR, I don't know. Keep in-camera NR down, always. And don't shoot Cine-D if you're going to use high ISO's. The noise it produces is very ugly. Use Natural or Standard profiles for less awful noise.
In your case, the only solution is to get faster lenses or lights to brighten the scene. I find that even as low as 1600 ISO, the G7 does an awful job in the underexposed areas. A little tip: maybe it's not a great help for you, as you're shooting people dancing, but if there's little movement in your shot, using a shutter speed of 1/40 instead of 1/50 doesn't hurt.
Hi Guy, thanks a lot for the fast answer.
The mark I version of the Mitakon Lens Turbo speedbooster is still sold new in my country (Portugal), so that's alright. I believe the mark II has already come, and is supposed to be for Nikon G lenses, so I assume it will allow for aperture control in lenses without an aperture ring. Here's the link, if this is what you were talking about: http://www.zyoptics.net/product/zhongyi-lens-turbo-adapters-for-micro-fo...
Anyway, I thank you for your suggestion, but I really can't afford it. I have found the used Nikon 28-85mm f/3.5 - 4.5 I was talking about in a local store for 100 $ with a Nikon to M4/3 adapter, so for the price of the Tamron I could buy the Nikon and the Speedbooster, and still have money enough to buy myself a good audio recorder.
I was just wondering if there was any possibility of incompatibilities between Nikon F lenses and Nikon AI/ AI-S lenses, or if it's all the same mount. But if you say that this lens should work fine with the Mitakon Lens Turbo (version 1), I will pick it right up this week.
Guy's answer is spot on. The GH3 and the G7 are so close, that the only way to decide is to figure out, within their differences, which is right for you.
I bought a G7 in August, and haven't regretted it for a single day. 30 min. time limit? I wish it wasn't there, but I know it is, so I just take care with it. No audio monitoring? Sucks, but it has audio levels on the screen and a decent limiter. Anyway, I prefer dual sound. Plastic body? Sucks, but I bought a cheap rain cover - not perfect, but works in light rain. But for the work I do - fiction shorts, mainly - the 1080p 4:4:4 (downsized from 4:2:0 4K footage) makes up for all that.
Between those two, there really is no one right answer.