It would be helpful if your project was in front of me. However, I will provide some general comments.
1. The RCA cables should be running from your head unit to the amp. The amp should have inputs to plug the RCA cables into.
2. The 4 RCA cables terminating into 2, would suggest that you have a 2 channel amp. A 4 channel amp would have inputs for 4 RCA cables.
3. Yes to: Head unit RCA's - OUTPUT to amp INPUT.
4. Difficult to diagnose the speaker wires going to the head unit and amplifier. There are some wiring harnesses that allow you to maintain your factory wiring, while adding an aftermarket amplifier. Not sure if this is your situation. Generally, the speakers are powered by the aftermarket amplifier. However, if you have the setup I described above, you may have some intermingling of the wiring. Again, difficult to say if the setup is not in front of me.
5. As it relates to your amplifier staying on, you need to find another source to connect your turn-on wire to. Generally, the wiring harness to your radio will have this turn-on wire. When you cut off your radio, the turn-on wire stop sending current to the amplifier tur-on. You may have the turn-on wire connected to the wrong wire coming from your head unit, Sounds like it is connected to the constant wire that powers your clock and memory functions. Look for a schematic for your radio to help identify the turn-on wire. If that is too difficult, use the accessory wire coming from your ignition switch. It cuts off and on with your ignition key.
If I have totally confused you I apologize. Maybe some other members can lend a hand. Good luck.