Okay, that's good: we know it's not the monitor.
The main things that can keep the PC from even displaying video output - the ones that you haven't ruled out yet, at least - are motherboard and CPU related. It's also possible both sticks of RAM are bad or that the PSU isn't providing enough power or is damaged in some way (still looks to be providing some power, though), but those are less likely at the moment. You'd have to have a PSU tester or multimeter or spare PSU to rule out the PSU entirely, but given that it's providing some power to the board I rule out the other things first.
It's too bad they didn't include a PC speaker with the motherboard as it's a cheap component, but oh well.
If you have one coming in, once you connect it and you do get beeps you can look them up based on the provider of the UEFI/BIOS, e.g. AMI (it should say somewhere in BIOS who the provider is, but I'm guessing it's AMI based on past ASRock mobos). A quick search of the motherboard doesn't show they list the beep codes in there, which is odd and annoying, but there are websites (like this one) that list what a BIOS provider's codes mean. Beep codes can't diagnose every issue, but it's a very useful tool specifically for situations like this.
Also, a careful rebuild outside of the case (putting mobo on an antistatic mat or the motherboard box) will eliminate shorts from the case as a cause, and if you *carefully* (don't want to bend pins in the process!) remove the CPU from the socket you should check for bent or broken pins, which you'll need to do anyway if it is the motherboard that's bad, as the manufacturer (ASRock) can deny a return if there's physical damage to the board.
