A:AnswerYes the AF-S NIKKOR 50mm f/1.8G Lens would be good for low light photography. The aperture can open pretty wide at 1.8 which will help you get as much light in as possible.
A:AnswerYes, the AF-S NIKKOR 50mm f/1.8G Lens will autofocus on the Nikon D5000. All AF-S Lenses with G and E type apertures will autofocus with the D5000. For more information please contact Nikon Customer Support.
A:AnswerYes the 50mm will autofocus on your body. It sounds as though either your 35mm is faulty or your focusing point of your 3400 isn't being placed where you think it is. Prime lens can be a little tricky. I would take it to a nearby camera shop and have them test it to see if it needs repairing.
A:AnswerYes, i use this lens with my Nikon d5500 and it works well. But the equivalent of this would be 75mm as this is an FX lens and d5500 is a DX format (might be hard when you use it in small spaces due to the crop factor). Nikkor 35mm f1.8 us available as well and that is already a DX format so you might want to consider that as well :)