So here's the results of my research:
It appears that COMBATHARDNESS does rise as you attack enemies, but very slowly. Being attacked, especially suffering severe injury such as loss of a limb, increases it fastest, but cumulative minor damage does build it up gradually as well. The level rises at a reasonable speed overall. Assuming you play it relatively safe in combat and don't suffer major trauma at any point, you should reach level 100 in COMBATHARDNESS at around the same time you reach Legendary level in your main weapon skill (probably). If you DO suffer major trauma, though, all bets are off. It is technically possible to go from 0 to 100 in one battle if you are beaten to a bloody unconscious pulp and somehow manage to survive. In practice, though, this is extremely unlikely to happen without manipulating events in the Arena.
Until your first fight, you do not have any COMBATHARDNESS at all and effects that require it will not activate, even if they only require you to be at level 0. I guess the counter only initiates after you get your first bruise.
Only physical combat with an enemy raises COMBATHARDNESS. Syndrome effects, jumping off of cliffs, being lit on fire and nearly drowning do not affect it. So no cheating your way to level 100 by carefully measuring the highest drop you can survive. This also means that melee fighters will level up much faster than those that fight using long-range interactions.
As far as I can tell, watching your allies die has no effect on COMBATHARDNESS in Adventure Mode. I guess adventurers don't have much in the way of personal relationships. As watching allies die is a major source of trauma in Fortress mode, the aforementioned information on usual level-up rate does not apply in Fortress mode.
You can connect an ability to a COMBATHARDNESS level, but reaching the level in which you can use that ability does not trigger a 'You have gained the power to X' message. You can also connect a body transformation to a COMBATHARDNESS level. For all effects including new abilities and transformations, you can set a level at which the effect begins and ends, meaning that you can make multiple levels of a special ability or multiple transformation levels as you level up.