1) UNIX used to be a closed-source operating system, yes, made by Bell and AT&T.
UNIX is no longer an operating system, however. That particular kind is no longer sold, and the name is now controlled by a different group which maintains a paper standard -- defining what features and utilities a UNIX operating system is supposed to have without getting too specific about how it works internally. If you follow these papers, and certify with them, you can call your operating system UNIX.
These days, there are open UNIXes(the many kinds of BSD, some kinds of Solaris), closed UNIXes(AIX), and everything inbetween.
2) Linux is not UNIX. Linux and the GNU utilities were actually made in a spirit of competition with UNIX. Same with HURD -- HURD was actually the "official" GNU kernel, Linux was an upstart project which appeared out of nowhere and overtook it.
It's nothing like ReactOS either, which can run Windows programs natively -- you couldn't run HPUX executables, AIX executables, or SunOS executables on Linux natively. It wouldn't make much sense to even try, these proprietary UNIXes are designed to run on their own proprietary machines. For that matter, Linux on ARM is not compatible with Linux on x86!
What different UNIXes have in common is
source compatibility -- you can't expect to haul a program from an alien architecture and expect it to run, but you can hope to build it from source on some UNIX's own compiler and get the same effect. This is the sense in which different UNIX and UNIX-likes are supposed to be compatible. They have the same kernel features and programming language "construction kits". This is also how Linux has managed to spread to such a bewildering variety of architectures from supercomputers to set-top boxes.
3) Linux is a kernel. Linux plus the GNU utilities makes a complete UNIX-like operating system. HURD or MACH plus the GNU utilities also makes a complete UNIX-like operating system. MACH plus the BSD utilities could make a genuine UNIX-certified operating system. Different Linux distributions are the Linux kernel plus different userland utilities.
The thing is, Linux and GNU weren't made to be a UNIX -- they were made in direct competition with it. GNU even stands for "GNU's
not UNIX". This stems right from the bad old days when a license for the UNIX source could set you back a cool hundred grand in 1980 dollars... It eschews the UNIX name for legal reasons but is very similar. It matters less these days, now that AT&T doesn't control the brand and there's many open alternatives. I hope GNU will forget the old feud and get things UNIX-certified someday.