How can I test the power supply so that I can be confident that it's up to specifications? For example, that it's not a cheap counterfeit with an OEM label.
You can't, not easily. All you can do are very basic tests like, if the supply is rated for 500W, pull 500W from it and see what happens - if it goes into protection, explodes, output voltage is 25% below spec... that pretty much settles it. But more elaborate tests like, how reliable is it over time, whether it's genuine, etc. require extensive experience. Testing products thoroughly is hard, a whole field of engineering in itself.
Which is why it is often worth it to pay more and go to a reputable manufacturer (and distributor!) rather than muck around on eBay, unless you know exactly what you are doing and have the experience to tell apart gems in the rough from garbage.
Moreover how can I test the "health" of a battery, and understand the relevant characteristics that are required to power the laptop computer?
Same applies for batteries. If the label says 2500mAh and you only get 1500mAh out of it after a full charge-discharge cycle (with the proper C), it's either toast or the label is a lie. But there can be many other things wrong with it which aren't so easy to detect - high internal resistance, internal shorts, unsafe protection circuitry, etc
How can I replace bad cells in the battery?
Cells in a battery pack should be replaced with some that are as similar as possible to those remaining - same chemistry, same voltage (of course), same capacity, same current rating, ideally same manufacturer. And same state of charge.
How can I safely teardown a PSU to determine why it's faulty
Best not to, if you're not sure of your skills. But if you must, the main danger is the capacitors. Make sure you understand how to ensure that they're discharged and stay discharged.