Simon, your guesses were pretty good
As you say, the second transistor in a Darlington pair cannot saturate. Even if Q1 was able to saturate to 0V across its collector-emitter, you need to have ~0.6V across the B-E junction of Q2. In practice, Q1 is likely to saturate to around 0.3V, so the lowest saturation voltage you should expect from a Darlington pair is around 1 volt (~0.3 plus ~0.6). Approximately...
Q2 is not damaged by too much base current because too much base current would (in theory) turn off Q2 by starving it of base current (by reducing Vbe). To make analysis easier, replace Q1 with a short, so the base of Q2 is connected to the collector. In that situation, the total current entering will divide - a little goes into the BE junction, the rest goes via the CE junction. Funnily enough, the ratio of this current division will be Hfe (or Beta if you prefer).
Funnily enough, I've met plenty of really quite experienced engineers who don't appreciate these facts, and assume that a Darlington will saturate to ~0.3V like a singe transistor...
If you want to think of a Darlington as being a single "transistor" - not a bad idea sometimes - then you just need to remember that the Vbe is now ~1.2V, and that the Hfe is the product of the individual transistors. The higher saturation voltage is rarely a big deal in linear applications (e.g. audio amplifier), but if using them to switch loads, then yes, they might run warmer than you expect.
If working quickly, a resistor across the BE junction of Q2 helps. Values vary according to application - perhaps 100 ohms might be a good start...