Well, we'd really need to see some screenshots at the very least along with better descriptions of the issues. I'm a fan of Google Fonts...for the web, but I don't get why you'd use them instead of Adobe's offerings.
I'm with you on using Adobe fonts but this was a corporate (government office) decision (meaning no one who knows better was asked).
We are now restricted to 3 google fonts (not a typeface with a variety of fonts but just a single font) which means we're going to have to get very creative with our software in order to make our work look as good as it should.
I've done some testing with InDesign's Justification settings and have managed to get a reasonable result but I still suspect that Google fonts have little, if any, metrics built into them.
I've set Justification settings to more extreme glyph scaling and have increased word spacing. The results aren't great but better. It's just not as polished as the Myriad Pro typeface we used previously.