Really, it is done in the inverse way.
We have different Unix versions: two main families: SystemV and BSD, different manufacturers, and so there was a need to standardize. US government wanted also standardized programs, so POSIX (version 1) was created, by standardizing OS interfaces (a step further than just C standard).
Windows NT is also POSIX (version 1) compatible, just because government wanted standardized tools. So POSIX was designed very very broad.
Then with time, there were need to standardize some more Unix (and similar) systems. Not as just one system, one API, but as common API, and so programs (e.g. GUI libraries or databases) could eventually use extension, but also make sure that program that follow the standard works on compatible system.
This was SUS (Single Unix Specification). This required a UNIX like system (unlike POSIX 1).
Then POSIX became not so important: application that in theory could work on all POSIX systems didn't really work on POSIX Windows.
So the new version of POSIX merged old POSIX plus SUS plus new useful function missing in SUS.
Now Linux is important, so Linux implementations (e.g. glibc) is taken into account when updating POSIX. You will see in the mailing list, that POSIX is defined by "vendors" of different Unix and similar systems.
So, it is not that operating systems extend POSIX, it is just that POSIX takes the most useful and standard options from different OS. It creates new interfaces just when existing interfaces are so incompatible, that by standardizing, it will break existing programs.
For the "second" question: when you develop a new operating system, you choose what way to go. Usually it is just derivation and fork (and distributions): again from the two Unix families, of just deriving Linuxes from RedHat or Debian). Sometime system is build from scratch, because of the design. Kernel provides most of system calls, so e.g. glibc needs a lot of systemcall (given by kernel) implemented in a similar way as POSIX. Glibc is not complete. Note: early Linux distributions used other libraries. GLibc was also written from scratch.