Can I use Software Center purchases on other distros or other OSes?

Maybe.

That type of usage is unsupported. There may be ways to make the packages work on different releases of Ubuntu, or even on other distros, but the details are up to you. Support for third-party apps is currently limited only to installing the packages on the exact distro they were built for. So, if you bought an app for Ubuntu 11.04, you can expect it to work on Ubuntu 11.04 but not on 10.10 or 11.10 or Fedora or any other Linux distros. It is, however, likely (but not guaranteed) to continue working on newer distros such as 11.10 and 12.04.

As of August 2011, Canonical is investigating ways to make commercial apps work on newer Ubuntu releases than they were created for, but nothing is decided yet.