Is OSX a POSIX OS?
Yes. POSIX is a group of standards that determine a portable API for Unix-like operating systems. Mac OSX is Unix-based (and has been certified as such), and in accordance with this is POSIX compliant. POSIX guarantees that certain system calls will be available.
Essentially, Mac satisfies the API required to be POSIX compliant, which makes it a POSIX OS.
All versions of Linux are not POSIX-compliant. Kernel versions prior to 2.6 were not compliant, and today Linux isn't officially POSIX-compliant because they haven't gone out of their way to get certified (which will likely never happen). Regardless, Linux can be treated as a POSIX system for almost all intents and purposes.
Yes, OS X is based on Darwin BSD, and since 10.5 (Leopard - 18-May-2007) all Intel/AMD versions have been officially certified as compliant with the Unix 03 / POSIX standard eg.
No, it is not. MacOS is missing a whole bunch of features of POSIX, like clock_nanosleep(). It might be compliant with a subset of POSIX or with a really really old version of POSIX, but it's definitely not compliant with POSIX.1-2017. See http://pubs.opengroup.org/onlinepubs/9699919799/functions/clock_nanosleep.html.