Is it possible? A PNG is generally less in size than an SVG. Sometimes I need small sized images in sites like Launchpad.
Note: When I simply save as PNG, the background remains white, not transparent.
That's actually Inkscape's default.
Open File → Document Properties and click the background colour. Make sure the alpha channel is 0. As so:
Go to File → Export Bitmap (Shift+Ctrl+E) and save a file with a .png extension.
Shortcut, at least in MS Windows, is (Shift+Ctrl+E).