I have forgotten the beep codes. But yes, I see no reason why you can't pair rx580 with core 2 duo. If the old gts works try it out first to rule out graphics card problems, then the core 2 duo if the prob still exists. Ram checking will be a prob. As you only have 1 stick. If you can manage another (borrow) you can rule out ram if that stick works and pc boots properly. In my limited amount of knowledge.
According to this link concerning MSI Boot up Beeps : BIOS beep codes. | MSI HQ User-to-User FAQ it is either your GPU card or Monitor that is having problems while booting up. I believe you must have the AWARD BIOS on your Motherboard. This is the only BIOS that indicates the sequence of beeps you mentioned. Here is from the link I posted first:
1 short - System boot successfull
2 short - CMOS setting error
1 long 1 short - DRAM or M/B error
1 long 2 short - Monitor or display card error
1 long 3 short - Keyboard error
1 long 9 short - BIOS ROM error
*repeated beeps - DRAM error
*repeated short beeps - Power error
*Repeated high & low beeps - Processor is damaged/Overheated
Try reinstalling your GPU card including making sure you have both GPU Power connectors connected to the card. According to this website, You have a sufficient PSU to run your GPU card. The minimum required PSU is 550 watts for total system requirement using a RX580 card.
I would check your monitor to be sure it is connected correctly and powered on.