How do I test that it actually is working as intended (resistance not so high as to fail to protect the computer parts on it from ESD, and not so low as to fail to protect the human from electric shocks)?
The computer's enclosure is a Faraday cage and works way better at protecting the parts inside than any ESD mat would. Keep the computer assembled and you won't have to worry about any of it.
If you're servicing the computer, then you don't need a mat either.
Have an anti-static wrist strap with a cord with a crocodile clamp at the end. You'd clamp it to the case of the computer.
Only place computer parts on the metal parts of the chassis while doing the servicing.
I presume that normally you're not swapping out motherboards, but maybe addding RAM or storage. For such occasional use, an antistatic mat is unnecessary, and a wrist strap will offer sensible protection to the parts from being touched by you. No other protection is necessary. A mat without a wrist strap is just about useless.
I've been messing about with my PCs for almost 4 decades and I've never used a an ESD mat, and I had no failures that could be reasonably attributed to ESD. I keep one hand or a forearm resting on the chassis, and that's enough to dissipate any charges my feet or my bum would be accumulating from rubbing the floor/chair. If I feel too lazy to pay attention to my arm posture, or if I'm doing stuff with sensitive T&M gear, then I put on a wrist strap and keep it connected to the rack or the chassis. There are some expensive components in RF test gear that are extremely ESD sensitive and can't include ESD protection elements while retaining their primary function (e.g. RF mixers).
Everyday PC stuff isn't even nearly in the same category, and every semiconductor component on the boards in a PC has individual ESD protection on its pins. The subassemblies thus have redundant ESD current sinks on most circuits. All you need then is not to have your body at a potential far above or below the chassis - and the wrist strap does that admirably.