A:AnswerYes it will work but just be aware it is meant for full frame so you'll be getting closer to a 50mm field of view on your D5300 (which is DX/crop sensor) compared to a 35mm field of view that a full frame camera would get. To be honest, a 35mm on DX is a super versatile field of view.
A:AnswerYes, it should work. However, note that it will crop the sensor as the Sigma Lens is a FX lens and the D3500 is a DX body. So instead of getting a 35mm crop, the image will look more like a 50mm field of view (52.5mm to be exact).
A:AnswerYes, the Sigma 35mm f/1.4 is designed to work on both APS-C / DX and full-frame / FX sensors. I’ve been using it for sometime on both my D750 and D7500, the images are amazing! The price deference between this and the Nikon version was the main reason I went with Sigma. The image quality is practically the same as the Nikon ( In my opinion i think the 35mm sigma does a better job with its focusing ).
Hope that helps!