You are going to want to take some photos, pad your post count, then post the photos of what you have.
CL3 means that it is rated for installation behind the wall (fire code).
If you have a cable, it will be terminated with a HDMI connection on it, it will not be a 'bare wire'. So, my feeling is that it is not a HDMI cable, but is more likely some sort of speaker wire.
1. If you need to connect speaker wire to your A/V receiver, then you connect it as described in the owner's manual for your A/V receiver.
2. You shouldn't need a connector, but if you DO want an easier connection to your A/V receiver, then pick up banana plugs which are very easy to connect. You will also need something to strip the wires. You can use household scissors or a sharp knife (carefully!).
3. You don't power speakers with an HDMI cable. You can extend speaker cables with wire nuts or in-line splices easily.
4. Yes, it is odd that if there are four speakers there is only one cable. If there are TWO speakers, then there may be a four conductor cable (red/black/white/green). That's when photos help.
5. You will want to ask questions and learn more and use Google a great deal if you want to follow up and get more specific answers to your questions.
6. Maybe, but it depends on your technical ability and willingness to learn.