1) Can someone make it very clear just what the relationship of OpenIndiana to IllumOS is?
IllumOS is the base operating system, much like Linux, except that it comes with a full user land too.
2) How exactly does NexentaOS fit in? And NexentaStor? And StormOS? And SmartOS?
Those are all distributions of Illumos. All of them contribute to Illumos and build on top of it by providing their own packages/packaging systems and system that run on top of Illumos. Think of them like Ubuntu/CentOS/Debian to Linux.
3) At least several of those I mentioned are open source/free, and I believe there are others. Why so many forks? Which one looks like the leader?
Illumos is the "leader", and the base operating system that all of those products use (AFAIK). Each of them have different options/features. NexentaStor for example is built to be a ZFS based storage appliance solution, SmartOS is for datacenters/virtualisation and things of that nature. They each bring something unique to the table. Each of them is built by a different company that offers different types of support.
The product formerly (freely) available as OpenSolaris had a lot to recommend it. FreeBSD has been playing catchup and has come a long way, but is still lacking in various ways. Linux is an excellent product, but glaring probems exist in the direction it is going, and I don't see it ever coming close to matching the OpenSolaris feature set in my lifetime.
OpenSolaris is still around, just with the name changed to OpenIndiana. OpenSolaris after a pkg mirror location upgrade was readily renamed to OpenIndiana, and this was the upgrade path that I took personally.
Hope this helps clarify things a little.