Yes it does extremely well and has different options for you to take low light images like Twilight mode. You can see comparisons to other big boy DSLRs to see how clean and detailed the output is at high ISO. One of the reasons I bought it.
Yes. It has the same sensor as the Nikon d7000 and you can shoot high ISo. Have to stay at the lower end of your zoom to keep the aperture (f stop) as wide open as the lens supports. Decent optical stabilization so can get good low light shots. Awesome camera!
It depends on whether your point of reference is a compact camera or a full-frame DSLR. If you want to compare it to the low light performance of a compact camera or pro compact camera, the answer is yes, the NEX-6 (with the 16-50mm kit lens) performs well in low light. This is not surprising because it has a much bigger sensor (APS-C). It is probably on par with many good DSLR cameras as well in terms of low light performance. Cameras with a full-frame sensor (such as the NEX-7 or the RX1) will still have an advantage, though. Also, keep in mind that a big factor will be the light-fastness of the lens you choose.
Yes, it does shoot well in low light but you need to make sure to turn on the focus assist lamp on on the body. The range on the assist lamp is good for the kit lens but will become less useful if your subject is far away.
Yes, very well. This morning I made interior photographs of my neighbors' newly redecorated rooms, which were illuminated only by daylight through windows. Set for Intelligent Auto, the camera made quick multiple exposures of each shot, processed them for a few seconds, and then displayed a gorgeous view with high dynamic range (i.e., details in both shadows and bright areas). This evening, just as dusk faded to night, I made photographs of the exterior of a building with light shining through windows plus paths lighted by street lamps. No matter what the setting on the camera - Intelligent Auto or Hand-Held Twilight or Night Scene - the exposures were excellent and the focus correct. The camera is a delight to use and works very well in low light - plus that cute little pop-up flash is always at the ready and in my so-far limited experience does the job for interiors, even when the lens is at its widest setting