It depends on your cassette deck and PC. The usual Walkman type cassette players have a standard 3.5mm jack, and the usual PC audio card has the same. If that is what you have, then no, this cable won't work. You would need a simple 3.5mm-3.5mm cable (http://www.amazon.com/Cables-Unlimited-AUD-1100-06-Stereo-Cable/dp/B000SE6IV8). If you're using a stereo to play your cassettes and it has RCA outputs, then this is probably the cable you want.
The mini plug should be the right size for your PC or a headphone connection. The Phono red/white would of course connect to your stereo or other amplified device. I use this to connect my Laptop to a AUX on a bookshelf stereo system. I also use the VGA from the laptop to my 40" tv but then that's another story!
Theoretically, HDMI should work much better than this Audio Y Cable Splitter in your case. I got the Audio Y Cable Splitter for my soundbar & TV, it works as it's intended w/ my TV and soundbar. I hope it'll work w/ yours as well.
What do you mean it doesn't have an audio jack on the tv? I've never seen a tv that didn't have audio output jacks. There should be a section on the rear of your tv that says output and has red & white RCA jacks that are used to wire your tv to a surround sound system. So if you have these output jacks, you could use this cable. However I would personally get a male RCA to male RCA cable to connect to you're logitech z506, since it has both RCA and mini stereo 3.5mm jacks.