Is there a free program to join and cut AVI movie files?

AVITools is trialware, but all it does is nag you and limit you to one job at a time.

MPEG Streamclip is another free Mac app that, despite the name, can mux together AVI, MPEG, MKV, and other sorts of video files into one big file.

And there's always good old FFMPEG, but it isn't quite as user-friendly as the above.

You can also use the concat demuxer with ffmpeg:

ffmpeg -f concat -i <(for f in $PWD/*.avi;do echo "file '$f'";done) -c copy output.avi

-c copy disables re-encoding the streams.

You can install ffmpeg with brew install ffmpeg after installing Homebrew.

See https://trac.ffmpeg.org/wiki/How%20to%20concatenate%20%28join,%20merge%29%20media%20files for more information.