The short answer is, thankfully, yes to all of the above. Including “which is best” unfortunately. There is no real magic involved to these things, despite what we may see on the advertising, and the mechanism is relatively simple. While the claims of special healing properties of different ingredients are fairly debunked, almost every functional topical pain reliever that does not have a direct medication like lidocaine works under a similar idea in that they will usually contain a light sensory irritant. This does not necessary mean that you are “irritating” the skin, but the chemicals involved will perform some sort of function that acts on some of our sensory receptors in the skin. Chemicals that evaporate quickly will produce a rapid “cooling” feeling like ice being applied, while others may feel more like a warming sensation if they effect the skin directly. Regardless, the input to the brain is similar: a novel and non-threatening stimulus is being input over the target area, and the brain can only concentrate on so much input at once. A constant background pain will be downgraded in the brain so it can focus on this new stimulus, and if it is something pleasant like warming or cooling, it actually may decrease the threat level held in the brain regarding that area, which can even decrease pain after the direct chemical effects have worn off, allowing the brain to reset its perception of how severe a threat to an area is. This takes advantage of the fact that pain is not directly correlated to damage (which is why a paper cut can feel like the end of the world but you might be able to play on a terrible sprain) but to the brain’s perception of how severely effected by danger an area is. If we can calm that down with a topical treatment, that’s great! Use it!
So are they safe? Well, short of some specific allergies you may have specifically, in general all of these chemical are chosen specifically because they are not at all dangerous, just like we talked about above. It is always a great idea to consult your physician if you are not sure about a possibility of allergic reaction, especially if you happen to have many allergies, but otherwise you would be very unlikely to see anything adverse if used as advised (that said, more sensitive areas of the skin like around the eyes may get far more irritated by these chemicals, and those areas should be avoided).