Unfortunately he had to replace it because it didn't manage to draw enough power for the lights.
Some dimmers are "not happy" with low power loads... A little night-light may not dim properly. Some LED light bulbs are not dimmable, or they may not draw enough power for the dimmer to work properly. Most fluorescent bulbs don't work with a regular dimmer either.
Also, the dimmable LED may not follow the same dimming-curve... My girlfriend has 3 regular 75W incandescent floods, plus one "equivalent" LED on the same circuit. With dimmer at minimum the incandescent lights are barely glowing and the LED is quite a bit brighter, but it is definitely dim. (This is an Insteon home automation dimmer and it dims in "steps", 16 steps I think, so there is a minimum before it goes completely off.)
- By my understanding, the dimmer itself doesn't draw much current; rather, it's the bulbs (load) that draws current through the dimmer (triac and resistors). Correct?
Correct. The TRIAC [u]switches on & off twice per cycle[/u] (when it's dimming) so theoretically, it's just like a light switch turning on & off very-quickly without using any power itself. However it's not 100% efficient and that's why they sometimes get warm.
So, it's wasting a little power but you're still not using as much power as the light at full-brightness. ...But in the winter the heat energy isn't wasted because helps to warm your house, taking a tiny-tiny load off your heater. 
It the summer your air conditioning (if you have AC) has to work a tiny-tiny bit harder so the energy is double-wasted.
- Very often a standard household dimmer tops out around 10 amp, right?
I'd say usually less. Here in the U.S. most of them are rated at about 600W. (about 5 Amps at 120V).
- If this dimmer reached it's breaking point at 10 amp, it would be weird, as 9 (for example) down-lights don't draw that much current. Yes?
It's probably going to burn-up. Semiconductors tend to die by shorting-out, so it might just stay-on at full-brightness.
- Having a regular household dimmer taking more than 15 amp is usually not necessary?
It depends on the load. I've only seen Wattage ratings. Current = Wattage/Voltage. 15A at 230V is 3450W so that would be an "industrial application". What's your 230V circuit rated for? Here in the U.S., we usually have 15 or 20A circuit breakers.
- Are fuses usually built into such dimmers, so that if they reach high currents they won't get warmer than they are supposed to get?
Probably not a regular fuse. There might be a thermal fuse. I've seen heaters & hair driers with a thermal fuse... If the fan stops working or the airflow otherwise stops the thermal fuse blows. In any case, if it's a "legitimate" dimmer with the required safety/regulatory approvals it shouldn't fail in a way that starts a fire, even if you overload it.
- For very basic household dimmers, only some resistors, a triac and an optocoupler are in metallic contact with the mains current. Correct?
Usually everything (except the grounded case) is connected to the power line.... Nothing is isolated (except the case).
If you build a dimmer with "wires coming out" for connection to the Arduino, the external low-voltage connections must be isolated. An "Arduino light dimmer" usually has TWO opto-isolators - One for the TRIAC-trigger and one for the phase detection. When I built a microcontroller dimmer I used the transformer from the power supply I built to isolate the phase detection.
It didn't make sense to me either. Seems like I didn't misunderstand this at all. I don't think the dimmer was very old, but I'm not sure, and I don't think it's due to the lamps.