I'm a little confused by the distinction. For example, if I use a framework that abstracts away pthreads from me, but still uses pthreads in its implementation of threading, is the application I write still POSIX-compliant?
POSIX compliance is much more common in consumer OSes nowadays, to the point that every major desktop and mobile platform is built on a POSIX-compliant kernel and runtime except for Microsoft's offerings. In that sense POSIX has taken over the non-Microsoft world, but now it's being used as a much lower-level compatibility layer for system-level frameworks rather than what it was originally designed for.
Even Windows has had multiple POSIX implementations over the years, they were just built as compatibility layers on top of the native APIs. So POSIX-compliant operating systems are almost universal at this point, but POSIX-compliant applications/daemons/etc. (all the programs that run on those POSIX OSes) are rarer than ever.
Yes. Part of this is because more operating systems are approaching POSIX-compliance; but the number of patches we need to port random Linux code to run on FreeBSD has dropped tremendously over the past 20 years.
Obviously not, since you can't go and compile it on another POSIX system -- you are "abstraction compliant" (or rather "abstraction dependent" now.