If you are using a raster drawing program like Gimp, yes, 300DPI at the exact size will work. Make sure you add 1/4 to 1/2" bleeds to the edges so your cuts don't have to be perfect (there will be a little extra going past the edge in case you make a mistake).
With regard to a transparent background... if you are printing to white paper, the background will be white. So you have to take your printing material into account. You don't have to have a PNG to have transparent data, but the point may be moot depending on what you're printing to. PSD, PDF, PNG all support transparency. Not sure what the native GMP format defaults to.
As for the shape in Gimp... do you have a scan of the existing image to work with to use as a tracing base? If so I could probably make the shape for you and save it in a format that will work in Gimp. I do this type of work all day.