CentOS? It's my understanding they do a good job of emulating RHEL, which itself focuses on stability and security ueber alles. The stuff won't be the most modern (COS 6.4 uses Linux 2.6.32), but IIRC it's well-maintained with security patches & has SELinux. You should be able to upgrade minor releases (6.3 to 6.4, say) easily, but upgrading between major releases won't work & will require a reinstall.
I haven't used it much (I'm a Debian/Ubuntu/Mint weenie) so I can't tell you about the state of its documentation.
CentOS is, bug for bug, RHEL, as long as you don't turn on the Extras repository. Scientific Linux deviates a little bit, but it's still mostly RHEL.
CentOS/RHEL 6 sort of use the 2.6.32 kernel. It's really a RH specific kernel with lots of backports from newer kernels, so it's not as stale as the number implies. The packages are older in CentOS/RHEL then other distros, but they get security updates regularly. RH also has a really good release engineering team.
Minor upgrades are easy. Just update like normal, and that's it. The normal way to upgrade a RHEL server is to replace it with a new server, so yeah, you'll need a backup and restoration plan to install the next major version. The good news is RHEL supports their OS for at least seven years, so you won't have to upgrade if you don't want to.
Documentation is pretty good. The official RedHat documents work for the clones as well, and CentOS/RHEL are very popular.
RHEL and the clones do things differently then Ubuntu or Debian, so there will be a little bit of a learning curve. Ubuntu kid of splits the middle between Debian and RHEL, in terms of the way things are structured.
Relatively low maintenance
Good updating policies. My ideal OS would be one that keeps all packages up to date, in a way that avoids breaking things, in a rolling release model. But, that's not really possible. Ubuntu fails here, because the LTS releases don't get good backports, and the non-LTS releases require frequent major upgrades that are to be treated as somewhat dangerous.
I'm thinking that FreeBSD and Solaris/Illumos (but which side of the fork, and which distro if the Illumos side?) are probably where I need to look here, but I'm certainly open to continuing on Linux if that's what's best for me, too.
Low maintenance kind of rules out FreeBSD. I really like FreeBSD, but you really need to have a dedicated support structure for it. Ports is awesome, but you're server will be spending time compiling stuff when new updates come out. There are the pkg tools which allow installation of precompiled packages, but the packages aren't guaranteed to be up to date.
Solaris/Illumos are alien worlds. They are very different then Linux or *BSD. They're great at what they do, but they're odd ducks.