The C13 (often referred to in industry as an IEC cable) is an absolute standard for computer PSU mains connections. Cables with 15A north american mains (NEMA 5/type B) male end* and IEC female end are thus ubiquitous. Note that the HP PDU, since it's intended for direct use with computer equipment, features female IEC connectors. So you need an IEC->IEC cable (commonly referred to as IEC Extension) rather than a mains->IEC cable to connect devices to this PDU. These cables are a little less common, but still cheap.
Note that the dryer receptacle you have now (probably a NEMA 14) will not be matched by the connector that comes with the PDU, since data centers almost universally use the NEMA L-series "twist-lok" connectors for mains service. So, you'll need to splice on a new cable, which you should be careful with since this will carry around 240v 20A (plus or minus a lot, depending on machine load) in the configuration you describe. In an enterprise environment any non-locking connector is considered a hazard (unreliable, unsafe), and I think this should be wisely carried to homes. What I'm saying is that it might be worth it to have a twist-lok connector installed.
(*) In the theatre industry we call these Edison plugs. I'm not sure how common the term is in IT. There are a lot of standards that define these plugs, NEMA 5 is cited the most.
Straight answer:
The only way to measure the true current flowing on the 12V rail is using a Clamp amperemeter, also called a Current clamp. To be specific, you need one that is able to measure DC currents, so one using Hall effect feedback loop, preferably with a digital display.
See https://en.wikipedia.org/wiki/Current_clamp
These are not cheap, but still affordable.
To measure the current, you select all the yellow wires from your ATX power connector or a PCIe 6 or 8 pin connector or a HDD power connector and clamp the clamp around them and around them only. These yellow wires are used for 12V. Similarly, red is 5V and orange is 3.3V. Black is GND.
Then you see the current on the display of the current clamp.
Done.
Further thoughts:
If you can't get a hold of a clamp amperemeter, then using a cheap power meter between the wall socket and the power cable will give you the total power used by the computer. The PSU efficiency ranges from between 70% to 95% based on the 80Plus rating of the PSU. In addition if you assume that 90% of that power goes over 12V rails, you'll be right for a modern system. So
wall_socket_power × psu_efficiency × 90% / 12V
is also a good guess for the currents going through the 12V rail.
Nowadays, most of CPU, memory and mainboard power comes from the 12V rail, as well as hard drive and external PCIe GPU power. Non-GPU PCIe cards typically use the 3.3V rail. The 5V rail is largely unused, except for HDD electronics in some cases. Your calculation is right for the peak case in this regard, including power conversion efficiency. 90% is optimistic, but possible. You should add 3W for each stick of memory and 5-10W for mainboard chipset, though.
Altogether, your PSU appears to be enough for normal usage and the only moment when it'll get anywhere near its full capacity is upon boot, when the drives are all spinning up at once and the CPU is not yet programmed for power-saving and runs full-throttle. If this becomes an issue, the drives can be configured to be spun up in a staggered manner by the OS.
If you're suspecting the PSU to be a reason of unstability, attaching an oscilloscope to the individual voltage rails and watching for voltage fluctuation under different loads is a much better indication of an insufficient or failing PSU than measuring the rail current.
Best Answer
Something like that ASUS board is no different that a PC Laptop. It's geared for low wattage and power consumption. The external power bricks are just like you would find in a laptop or their old media centers. Personally I own both their old media centers CS5111 and several laptops. Of all of the things that have failed on them, the PSU (power brick) is not one.