I upgraded from Windows 8.1 to Windows 10 and have not encountered any issues with GSAK (I'd say it works better under Windows 10 than it did under Windows 8.1).
I use Chrome as my browser of choice, so can't say much on that front. I have no issues with geocaching.com, mygeocachingprofile.com, Project-GC, Rumkin (encryption resources).
I think my wife's computer came with Windows 7 and it upgraded to Windows 10 with no issues at all. GSAK works great on it.
Windows 7 entered extended support on January 13, 2015; What that means is that you will get security updates until January 13, 2020.