I just downloaded the installation for Ubuntu, but it is a file format I don't recognize - .7x. It won't open in file-roller, I can't tar with zxvf, jxvf, or simple xvf.

Can anyone tell me what this 7x file is, and how I can extract the installation files from it?

It's a 7zip file. (7z, not 7x.) You can get 7zip to extract it, or simply download the .tar.gz.

sudo apt-get install 7zip
then
7z x houdinix.x.

No need to download 7zip, the archives are self-extracting:

chmod +x houdini-9.0.*.7z
./houdini-9.0.*.7z