Relay that detects 240VAC and switches 12V DC

You should be fine driving the ULN2803 directly from a 74HC595; the latter can source the current that the Darlington driver needs.

My problem is that I don't understand what kind of load a contactor is (e.g. how much it is inductive, does it give current surges, or kickbacks as a DC relay, etc.)

The coil voltage is 230/240V rated and consumes 2.7 VA when activated (ignoring inrush). The holding current is therefore about 11.7mA RMS. It's mainly inductive and this is "indicated" by the inrush VA requirement - this is the magnetic core saturating causing this 9.2 VA. The 9.2 VA will last for less than 100 milli seconds (in my opinion) and you probably wouldn't even see this if you activated the coil right at the peak of a positive or negative voltage on the AC. Yes coil inrush is worst at zero-crossing!

The solenoid inductance can be calculated from the holding current of ~12mA. This implies an AC impedance of 230/0.012 = 19.17 kohms and, at 50 Hz, this suggests an inductance of 62 henries (this is needed to be a powerful solenoid).

Given that you have "almost no experience in designing circuits" I would use a control relay to activate the contactor and I'd put a snubber across the contactor coil. The snubber will prevent large arcs when the control relay open-circuits.

Contactor coil energy is 12mA (squared) x 62/2 = ~5 mJ and a 220nF 250V AC snubber capacitor would restrict back-emf from the coil to about a 200 volts kick-back. You'd probably get away with 100nF. A series resistor ought to be included and 100 ohms is going to be about the right ball-park (calculate power requirements though).

So, 100nF will restrict the kick-back to about 316 volts peak. The 100 ohms is really there to restrict the snubber current when the relay contact activates the contactor coil.