If the background is white or transparent, and your printer is printing black, that would seem to be a printer problem, to me.
However, Inkscape is not strong with printing. You could try using another software to print....well, I mean the PNG.
For the SVG file you attached, you could put a white rectangle the same size as the page, behind everything. That might force it to print white?
To put an outline around the PNG that you shared, I can think of 2 ways.
First, since it's a PNG, Inkscape can't do much with it, because PNG is a raster format, and Inkscape is a vector editor. So one way to add the outline is to convert it to vector using Path menu > Trace Bitmap. Then you can add a stroke using Object menu > Fill and Stroke > Stroke Paint tab, and click the Flat Color button, which looks like this
The other way would be to not convert it to vector. Instead, you could use the Pen/Bezier tool to manually trace an outline, right on top of the imported PNG. This would probably take a little longer, since there would be a bit of a learning curve with the Pen tool.