[Troubleshooting] First no display, now breakboarding, getting constant beeps

I had followed a build guide here and got everything all set up but when I went to connect my monitor there was no BIOS or any error messages just a blank screen briefly before the monitor would go back to a low power mode (normal).

Then i found this thread:
Post System Boot video output troubleshooting checklist

I have been going through the checklist on there all the way up to the breakboarding/ listening for beeps steps. So I stripped it all apart and put the motherboard on a wooden desk on top of its own box. I now have plugged in only the 8-pin CPU power plug, the CPU cooler (stock), the 24-pin motherboard power plug, the power button plug, and the little speaker. When I go through the boot process I get rapid beeps (about 5-10 per second, continuously) until I turn the power switch back off. I consulted my manual and a few threads here after some quick google-fu, and apparently that means it is a "power issue."

This is my first build ever; built around a budget of about $300. Here is what I'm using right now:
AMD Athlon X4 860K 3.7GHz Quad-Core Processor (w/ stock CPU cooler)
Gigabyte GA-F2A68HM-DS2H Micro ATX FM2+ Motherboard (rev: 1.1)
Crucial Ballistix Sport XT 4GB (1 x 4GB) DDR3-1600 Memory
MSI Geforce GT 610 2GB Video Card (model #: N610GT-MD2GD3/LP)
Seagate Momentus 7200.4 320GB 2.5" 7200RPM Internal Hard Drive
Microsoft Windows 10 Home (64-bit)
500W Power Supply integrated in the Cooler Master TC102 ATX Mid Tower case
2x Cooler Master Sickleflow 120mm Fans
CD Drive from my last computer

What I'm wondering is if this means one of my parts came in DOA or if there is some type of compatibility issue between some of the parts. I still have access to the 250W power supply from my last computer that lasted me about 10 years until a hard drive crash (reason for new build). Can I use that to confirm if its the new integrated power supply? I also still have the mobo and CPU from the old computer but its all Intel stuff so I assume that's of no help here.

The threads here have been very helpful up to this point so I'm hoping there's some way to figure out whats wrong even if i have to send back some parts and get new ones. Any insight would be greatly appreciated.