Welcome to the forum!
The proper way to get a PNG with Inkscape, is using File menu > Export PNG. You should not use File menu > Save As cairo png, unless you specifically want that kind of file. Save As cairo png does not support transparency.
I don't have any trouble saving those files as PDF. Although for the blue one, I have to check "Rasterize filter effects" box, to preserve the blue filter. I'm using Inkscape version 0.92.3.
I notice you report using version 0.91, and it's possible you're seeing a bug which has since been fixed. Although I also notice you're a Mac user, which means you might not be able to upgrade.
When you say the image disappears, you mean in Inkscape, right? Are you able to open the PDF file using a PDF viewer (such as Adobe Reader)? Does the image appear there?
I seem to recall something about transparency and a bug with PDFs in Inkscape (in 0.91). If just clicking Rasterize Filter Effects doesn't fix it, you could try putting a solid white rectangle, the same size as the page size, behind the image.
Beyond that, I can't think of anything that might be causing this problem. But maybe someone else will have an idea.
(Oops, Moini posted while I was testing and typing)