Are packages within a Linux distro interchangable?

More often than not you can make it work, eventually.

The question is how much work you want to spend tracking down dependencies and fixing paths; You can even install RPMs from redhat into Debian's dpkg system, but it requires a bit of time to actually get everything set up and working (I've done so before, and there was actually less hairpulling than you'd think, but it's all dependent upon which packages you're trying to install).

The binary files are usually compatible (unless there's some breaking change or customization in the kernel API of the specific branch you're using), and the only real trick to making it work is getting the correct version of the necessary libraries installed and set up so that the binaries can find them. If the branches are close enough together (like Ubuntu and Debian), then I wouldn't even think twice about trying to install a debian .deb into an ubuntu system-- almost always it just works.

It depends.

Sometimes you can use Red Hat packages on Fedora and vice versa, but not always as it depends on the dependencies of the package and specific revisions. Centos should usually work on Red Hat and vice versa, as Centos is just Red Hat minus the branding.

Sometimes you can use Debian packages on Ubuntu, but not always.

In my experience though, installing packages from another distro is a sure path to madness and loss of hair. More often than not it will go bad, although that may be 6 months down the line when it comes back from the grave to get you.