How to adjust rear brakes on this bicycle?
It sounds like you've really done some good work! The pad material shouldn't be too soft. Soft enough that it creates good friction when pressed against the rim, but still rigid enough that it holds its shape while doing so. Keep in mind they will break in after using them for a few days or a week or too. The pads you have look like they should be just fine, although it's hard to tell from pics, but Kool Stop pads are my favorites and I know they work great, if you  chose to try something different.